(Mathematics) in The University of Michigan .... 4.1.1 Group action . .... by an affirmative answer to the following question: Does a wedge centered at a âgen-.
Jet Schemes and Truncated Wedge Schemes
by Cornelia O. Yuen
A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in The University of Michigan 2006
Doctoral Committee: Professor Karen E. Smith, Chair Professor Melvin Hochster Professor J. Tobias Stafford Associate Professor Mircea I. Mustat¸ˇa Associate Professor James P. Tappenden
c
Cornelia O. Yuen 2006 All Rights Reserved
To jason
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ACKNOWLEDGEMENTS
There are many people to whom I would like to express my gratitude for helping me complete this thesis. First, I must thank my advisor, Karen Smith, for her guidance, caring, and patience. She is an excellent role model, both in matheamtics and in life. It is my honor to be one of her students. Secondly, I am blessed to have the spiritual support and love of jason howald, especially through the many difficult times in the past year. I also have to thank him for his careful readings and comments on this thesis. I must also thank Mircea Mustat¸aˇ, my second reader, for his expertise and all the valuable discussions and suggestions along the way. I am also grateful to Mel Hochster for bringing me up through the process of graduate school, especially at the beginning when Karen was away. Hai Long Dao’s clever idea involving additivity of length and Trevor Arnold’s help with LATEX diagrams were invaluable. Finally, I must thank my parents for allowing and encouraging me to live my dream.
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TABLE OF CONTENTS
DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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CHAPTER I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 1.2 1.3 1.4
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2 3 3 5
II. Background and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2.1 2.2 2.3 2.4
Overview of jets and arcs . . A higher dimension analog of History . . . . . . . . . . . . Overview of results . . . . .
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44 44 45 45 49 51 56 58
V. Truncated Wedge Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5.1 5.2 5.3
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IV. Jet Schemes of Determinantal Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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16 20 35 37 38 39
Basics of determinantal varieties . . . . . . . . . . . . . 4.1.1 Group action . . . . . . . . . . . . . . . . . . 4.1.2 Singularities . . . . . . . . . . . . . . . . . . . Are jet schemes of determinantal varieties irreducible? . 4.2.1 Odd jet schemes are reducible . . . . . . . . . 4.2.2 Second jet scheme is reducible . . . . . . . . . 4.2.3 Varieties of matrices of rank at most one . . .
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III. Jet Schemes of Monomial Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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8 11 13 13
Reduced monomial hypersurfaces . . . . . . . . . . . . . . . . Primary components . . . . . . . . . . . . . . . . . . . . . . . Fat points on the affine line . . . . . . . . . . . . . . . . . . . . 3.3.1 Proof of conjecture when m < n . . . . . . . . . . . . 3.3.2 Proof of conjecture in the case when m ≡ −1 mod n 3.3.3 A lower bound in the case when m ≡ −2 mod n . . .
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3.1 3.2 3.3
Scheme structure . . . . . . . . . . Jet schemes as representing schemes Jet schemes of smooth varieties . . Jet schemes of singular schemes . .
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Definition and Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Truncated wedge schemes of local complete intersections . . . . . . . . . . .
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64 66 72
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74 75 79 80
VI. Open questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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6.1
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Truncated wedge schemes of monomial schemes 5.4.1 Monomial hypersurface case . . . . . . 5.4.2 Alternate point of view . . . . . . . . . 5.4.3 Multiplicity . . . . . . . . . . . . . . .
Problems arising from specific results . . . . . . 6.1.1 Jet schemes of monomial schemes . . . 6.1.2 Jet schemes of determinantal varieties 6.1.3 Truncated wedge schemes of monomial Additional classes of schemes to be studied . . .
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BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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CHAPTER I
Introduction
In this thesis, we present some results on jet schemes and a higher dimensional analog, truncated wedge schemes. Where as jet schemes of smooth varieties are well understood, this thesis addresses the question: Question I.1. Given a singular variety X, what can we say about the scheme structure of its jet schemes and truncated wedge schemes? For example, are they irreducible? If not, how many components do they have? What are their dimensions? What is the multiplicity along each of the components? We study this question specifically in the case of monomial schemes and determinantal varieties, working over an algebraically closed field k of characteristic zero. The reason for looking at these types of schemes is twofold: First, their extra combinatorial structure adds richness which makes the study of their jet schemes and truncated wedge schemes feasible. Second, these varieties are themselves of interest because they arise naturally in several contexts in algebraic geometry. For details on the importance of each of these basic types of varieties, please refer to the introductory remarks of Chapters III and IV. We are able to answer Question I.1 to varying extents in different situations; see Section 1.4 for a summary of specific results.
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2
1.1
Overview of jets and arcs
An arc on an algebraic variety X is an “infinitesimal curve” on it. Formally, this is a morphism, defined over k, from the curve germ scheme Spec k[[t]] into X. As is typical in algebraic geometry, this set carries the structure of a scheme, called the arc space of X and denoted by J∞ (X). An m-jet on X is a truncated arc on X, that is, a k-morphism Spec k[t]/(tm+1 ) → X. We can think of such a jet as the choice of a point x in X, together with the choice of a tangent direction, a second order tangent direction, and so forth up to an mth order tangent direction at x. The set of all m-jets on X also forms a scheme in a natural way (see Proposition II.2). This is the mth jet scheme Jm (X). In particular, J0 (X) recovers X and J1 (X) = {Spec k[t]/(t2 ) → X} = T X (the total tangent space of X), since a tangent vector on X is simply a 1-jet by definition [19, Ex. II.2.8]. m Note that there are natural “forgetful” maps πm−1 : Jm (X) → Jm−1 (X) induced
by truncation of jets Spec k[t]/(tm+1 ) → Spec k[t]/(tm ) → X. Taking the inverse limit of these jet schemes gives the arc space J∞ (X). Jet schemes are well understood for smooth varieties. All orders of tangent directions at any smooth point are unobstructed, so that when X is smooth, Jm (X) is easily seen to be an Am dim X -bundle over X (see Corollary II.7). However, jet
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schemes of singular varieties are much more complicated and little is understood about them, which is the starting point for this thesis.
1.2
A higher dimension analog of arcs and jets
A wedge is a higher dimensional analog of an arc. More precisely, a wedge is a k-morphism Spec k[[s, t]] → X, which again can be thought of as an “infinitesimal surface germ” on X. In this thesis, we introduce and study the analogous notion to jets: Definition I.2. Let X be a scheme of finite type over k. An m-wedge of X is a k-morphism Spec k[s, t]/(s, t)m+1 → X. As with m-jets, the collection of all m-wedges forms a scheme Wm (X) in a natural way, called the mth wedge scheme of X (see Section 5.1). Analogous to the situation with arcs, the set of all wedges also forms a scheme W∞ (X) = ← lim −Wm (X), called the wedge scheme of X.
1.3
History
The first serious study of arc schemes appears to have been undertaken by Nash in the 60’s [31]1 . Nash was interested in whether the singularities of a variety X could be reflected in its arc space. In particular, he showed that for each “essential” exceptional divisor2 of a resolution of singularities of X, there is an associated component of the preimage in J∞ (X) of the singular set of X under the natural projection J∞ (X) → X. Nash showed that this association from exceptional divisors to components is injective, and conjectured that it is also bijective. 1 His
paper was written in 1968, but was published only recently. the surface case, these are exactly the exceptional divisors appearing on a minimal resolution of singularities. In higher dimension, these are the exceptional divisors (valuations) that appear on every resolution of singularities. 2 In
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This conjecture spurred much interest among singularity and valuation theorists, leading to many partial answers to Nash’s conjecture. For example, Nash’s conjecture has been proved for minimal surface singularities [33], rational double points of type Dn [32], sandwiched surface singularities [26, 34], and toric varieties of any dimension [21]. However, Ishii and Koll´ar [21] recently found a counterexample to Nash’s conjecture in general. Interest in jet schemes and arc spaces was recently rejuvenated by Kontsevich’s striking introduction of motivic integration in his famous 1995 lecture at Orsay [24]. There he proved Batyrev’s conjecture that birationally equivalent smooth CalabiYau varieties have the same Hodge numbers. Batyrev himself had proven weaker statements using p-adic integration; Kontsevich, inspired by the analogy between the complete regular local rings k[[t]] and the p-adic integers Zp , introduced “motivic integration” for certain functions on the arc space of a smooth variety, which has turned out to be a powerful technique for proving results in birational geometry. Since then the theory has been further developed by Batyrev [1, 2], Denef and Loeser (who consider arcs on singular spaces) [8, 9, 11], and others. For good surveys on motivic integration with an emphasis on geometric applications, see [4, 6, 10]. Motivic zeta functions can also be defined analogously to the ususal arithmetic zeta functions counting the number of Fpn -points on a variety defined over Fp , see [27, 36] for surveys of motivic integration and motivic zeta functions. Inspired by these developments but in a somewhat different direction, Mustat¸ˇa later showed that jet schemes can detect subtle information about the singularities of a variety X. For example, he showed that a local complete intersection has rational singularities if and only if all its jet schemes are irreducible [29], and he also characterized log canonical pairs by means of dimensions of jet schemes [30]. Mustat¸ˇa
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and his collaborators developed this line of research further, proving for example, a formula for the log canonical threshold of a pair in terms of dimensions of jet schemes with de Fernex and Ein [7], and an interpretation of multiplier ideals in terms of jets with Ein and Lazarsfeld [12]. In a striking recent application, Ein, Mustat¸aˇ and Yasuda used jet schemes to prove the famous Inversion of adjunction conjecture for smooth varieties [14] and for normal locally complete intersection varieties [13]. The study of wedges was initiated by Lejeune-Jalabert in 1980 in an attack on Nash’s conjecture [26].
Her idea rested on the observation that any wedge
Spec k[[s, t]] → X can be precomposed with the natural map Spec k[[t]] → Spec k[[s, t]], dual to the map of rings sending t 7→ t and s 7→ t, producing an arc called the “center” of the wedge. She showed that Nash’s conjecture (for surfaces) could be settled by an affirmative answer to the following question: Does a wedge centered at a “general” arc on a normal surface singularity lift to its minimal resolution of singularities? Later, Reguera [34] considered this problem for wedges on higher dimensional varieties, and showed that a positive answer to Nash’s question is equivalent to a positive answer to this wedge extension problem.
1.4
Overview of results
The main results in this thesis are in Chapters III, IV and V. In Chapter III, we study jet schemes of monomial schemes. In this case, the jet schemes are known to be equidimensional and without embedded components, but usually are not reduced [17]. We thus investigate their structure further, giving for example a formula for the multiplicity of these schemes along each of their irreducible components. When the monomial scheme is a reduced monomial hypersurface (that is, the case of simple normal crossing divisor), the multiplicity is a multinomial
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coefficient (see Theorem III.1). When the monomial scheme is a multiple point in the affine line, the multiplicity appears to be a binomial coefficient (see Section 3.3). A much harder task is to explicitly describe the generators for the primary components of the jet schemes. We under take this project with some success for some of the components of the jet schemes of Spec k[x, y]/(xy) (see Section 3.2). In Chapter IV, we study the jet schemes of determinantal varieties. We show that these jet schemes are reducible in general (see Theorems IV.8 and IV.18). This is somewhat surprising in light of Mustat¸aˇ’s work indicating that jet schemes of mildly singular varieties are often irreducible [29]. We also find the number and dimensions of the irreducible components of the jet schemes of essentially all determinantal varieties of matrices of rank at most one (see Theorem IV.19). As an application, we show that the log canonical threshold of the pair (Ars , X) where X is the variety of r × s matrices of rank at most one is exactly rs/2 (see Corollary IV.21). In Chapter V, we develop the theory of truncated wedge schemes. We start by proving some basic properties; for example, truncated wedge schemes behave well under ´etale morphisms (see Proposition V.3) and truncated wedge schemes of a smooth scheme are affine bundles over the smooth scheme (see Corollary V.5). We also show that the first wedge scheme is the product of the first jet scheme with itself (see Proposition V.6). Furthermore, we prove an irreducibility criterion for truncated wedge schemes of a locally complete intersection variety (see Theorem V.8), analogous to a criterion of Mustat¸aˇ for jet schemes [29]. Also in Chapter V, we conduct a detailed study of the truncated wedge schemes of monomial schemes. We show that the reduced subscheme structure of the mth wedge scheme of any monomial scheme is itself a monomial scheme in a natural way. We give explicit generators of each of the minimal primes of any mth wedge scheme
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of a monomial hypersurface scheme (see Theorem V.11), analogous to [17]. Interestingly, unlike the situation for jet schemes, these truncated wedge schemes need not be equidimensional. They may even have embedded components (see Remark V.16). However, in one important way, it appears that truncated wedge schemes are nicer than jet schemes: it seems that the irreducible components of the truncated wedge schemes of a reduced monomial hypersurface all have multiplicity one. We show this is true for reduced monomial hypersurfaces in two or three variables (see Theorems V.17 and V.18). We also present further evidence of this conjecture based on Macaulay calculations.
CHAPTER II
Background and Preliminaries
In this chapter, we review the basic theory of jets and arcs. In particular, we explain why the set of all jets (or arcs) naturally form a scheme, and recall a few basic properties we will use. In Section 2.4, we present a framework for understanding jet schemes of a singular variety that will form the backbone of our approach to understanding jet schemes of determinantal varieties.
2.1
Scheme structure
Definition II.1. Let X be a scheme of finite type over k. An m-jet of X is an infinitesimal curve of order m; that is, a morphism over k Spec k[t]/(tm+1 ) → X. m The surjection k[t]/(tm+1 ) → k[t]/(tm ) induces a morphism πm−1 : Jm (X) →
Jm−1 (X), and composition gives a morphism πm : Jm (X) → J0 (X) = X. Furthermore, the association of Jm (X) to X is functorial: given a morphism X → Y , there is an obvious induced map of sets Jm (X) → Jm (Y ). An arc is a limit of m-jets, in other words, a k-morphism Spec k[[t]] → X. The arc space of X is J∞ (X) = lim ←−Jm (X).
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Proposition II.2. Let X be a scheme of finite type over k. The set of all m-jets, Jm (X), carries a natural structure of a scheme of finite type over k. Proof. First, if X = Ar , then an m-jet of X is equivalent to a k-algebra homomorphism φ : k[x1 , . . . , xr ] → k[t]/(tm+1 ), which is completely determined by where it sends the coordinates xi . Say (0)
(1)
(m)
φ(xi ) = xi + xi t + . . . + xi tm (j)
with xi ∈ k. Then the mth jet scheme Jm (X) can be identified with (0)
(m)
(m) Spec k[x1 , . . . , x1 , . . . , x(0) r , . . . , xr ]. m : Jm (X) → Jm−1 (X) are affine bundles (and so In particular, the morphisms πm−1
surjective), since they are induced by the inclusions (0)
(m−1)
k[x1 , . . . , x1
(0)
(m)
(m−1) (m) , . . . , x(0) ] ,→ k[x1 , . . . , x1 , . . . , x(0) r , . . . , xr r , . . . , xr ].
Therefore, Jm (X) is an Amr -bundle over X = Ar . Next, if X ⊆ Ar is a closed subscheme, then we illustrate that its jet scheme Jm (X) is a closed subscheme of Jm (Ar ): Say X = Spec k[x1 , . . . , xr ]/I. Then an m-jet of X is equivalent to a k-algebra homomorphism φ : k[x1 , . . . , xr ]/I → k[t]/(tm+1 ). The map φ is clearly determined by the images of the generators xi under φ, say (0)
φ(xi ) = xi
(1)
(m)
(j)
+ xi t + . . . + xi tm . However, there are constraints on the xi ’s.
Indeed, fixing a set of generators f1 , . . . , fn for the ideal I, we must have φ(fk ) = 0 for each k = 1, . . . , n, or that (2.1)
(0)
(m)
(m) m m+1 ). fk (x1 + . . . + x1 tm , . . . , x(0) r + . . . + xr t ) = 0 in k[t]/(t
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Rearranging (2.1) so as to collect like terms of powers of t, we have (0)
(1)
(m)
fk + fk t + . . . + fk t m (l)
(0)
(m)
(0)
(m)
where each fk ∈ k[x1 , . . . , x1 , . . . , xr , . . . , xr ]. Then Jm (X) is the subscheme (j)
(l)
of Spec k[xi ] = Jm (Ar ) defined by the polynomials fk , as k ranges from 1 to n and l ranges from 0 to m. It is not hard to check that this computation commutes with localization. Thus, this local construction of jet schemes can be patched together to give a scheme structure on the set of maps Spec k[t]/(tm+1 ) → X for any scheme X of finite type over k. By similar reasoning, we see that the collection of all arcs of X also carries a natural scheme structure for all k-schemes X, although this scheme is not finite type over k in general. When X ⊆ Ar is an affine scheme, we let Jm (X) denote the defining ideal of the jet scheme Jm (X) as a subscheme of Jm (Ar ) ∼ = Ar(m+1) . Also, we write Jm instead of Jm (X) when the scheme X is clear from the context. With that said, let us look at two examples of jet schemes. Example II.3. For any scheme X, J0 (X) ∼ = X and J1 (X) ∼ = T X, the total tangent space of X. Example II.4. Let X = Spec C[x, y]/(xy). Then J0 (X) = (x0 , y0 ) ⊂ C[x0 , y0 ] J1 (X) = (x0 y0 , x0 y1 + x1 y0 ) ⊂ C[x0 , x1 , y0 , y1 ] J2 (X) = (x0 y0 , x0 y1 + x1 y0 , x0 y2 + x1 y1 + x2 y0 ) ⊂ C[x0 , x1 , x2 , y0 , y1 , y2 ] In general, Jm (X) = (g0 , g1 , . . . , gm ) where gk =
P
xi yk−i is the coefficient of tk in
the product (x0 + x1 t + . . . + xm tm )(y0 + y1 t + . . . + ym tm ).
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Note that the truncation map π12 : J2 (X) → J1 (X) is not surjective. The surjectivity of π12 amounts to being able to fill in the dotted arrow below. γ / k[t]/(t2 ) NN OO NN NN '
k[x, y]/(xy)
k[t]/(t3 )
A 1-jet γ lying over the origin is defined by a map γ : x 7→ x1 t y 7→ y1 t for any values x1 and y1 from k. But a simple calculation shows that such a 1-jet lifts to a 2-jet γ e : x 7→ x1 t + x2 t2 y 7→ y1 t + y2 t2 if and only if x1 y1 = 0. This means there are tangent vectors of X at the origin that cannot be assigned an acceleration.
2.2
Jet schemes as representing schemes of a functor
Let us start with a brief introduction to viewing “schemes as functors”. Given a k-scheme X, we define the functor of points of X to be the functor FX : k-Schemes → Sets Z 7→ Homk (Z, X). A functor F : k-Schemes → Sets is said to be representable by a k-scheme if it is of the form FX for some k-scheme X. By Yoneda’s lemma [16, Lemma VI-1], the
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scheme X is unique if it exists. The set FX (Z) is called the set of Z-valued points of X. This generalizes natural intuition in the following sense: When X = Spec k[x1 , . . . , xr ]/(f1 , . . . , fd ) is an affine scheme and Z = Spec k, an element of Homk (Z, X) corresponds to a k-algebra map φ : k[x1 , . . . , xr ]/(f1 , . . . , fd ) → k. The map φ is determined by the images αi of xi , subject to the conditions φ(fl ) = 0 for l = 1, . . . , d. So φ corresponds to an r-tuple α = (α1 , . . . , αr ) ∈ k r such that α satisfies the equations φ(fl ) = 0 for all l. In this way, Homk (Spec k, X) is identified with the set of k-points of X. For general k-schemes X and Z, we call Homk (Z, X) the set of Z-valued points of X though Spec Z is not necessarily a point anymore. Remark II.5. By an improved statement of Yoneda’s lemma [16, Proposition VI-2], we may restrict ourselves to the category of affine schemes when looking at functor of points. We will see that the jet scheme Jm (X) of a scheme X has the characterizing property that it is the scheme representing the functor k-Schemes → Sets Z 7→ Homk (Z ×k Spec k[t]/(tm+1 ), X). In other words, Homk (Z ×k Spec k[t]/(tm+1 ), X) = Homk (Z, Jm (X)) for all k-schemes Z. The proof is very similar to that of an analogous statement for truncated wedge schemes, so we will prove the more general statement in Chapter V (see Proposition V.2). Note that when k is an algebraically closed field and Z = Spec k,
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the closed points of Jm (X) are in one-to-one correspondence with the set of maps Spec k[t]/(tm+1 ) → X; that is, the set of m-jets, just as expected.
2.3
Jet schemes of smooth varieties
In this section, we show that the jet schemes of a smooth variety are smooth. This is a consequence of the good behavior of jet schemes under ´etale morphisms: Proposition II.6. Let X → Y be an ´etale morphism of k-schemes of finite type. Then Jm (X) ∼ = Jm (Y ) ×Y X for all m. The proof of this result is nearly identical to the case of truncated wedge schemes, which is explained in Chapter V (see Proposition V.3). Corollary II.7. If X is a smooth variety over k of dimension n, then Jm (X) is an Amn -bundle over X. In particular, Jm (X) is smooth of dimension (m + 1)n. Proof. Since X is smooth over k, X is covered by an open affine cover {Ui }i with Ui → Vi ´etale, for some Vi open subset of An . Then Jm (Ui ) ∼ = Jm (Vi ) ×Vi Ui
(by Proposition II.6)
∼ = (Jm (An ) ×An Vi ) ×Vi Ui
(since an open immersion is ´etale)
∼ = Jm (An ) ×An Ui . Now Jm (An ) is an Amn -bundle over An implies that Jm (X) is an Amn -bundle over X.
2.4
Jet schemes of singular schemes
Let X be a reduced and irreducible scheme of finite type over k with singular set X sing . Then X is the disjoint union of X sing and its complement X reg . Considering
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the natural projection πm : Jm (X) → X, we see therefore that Jm (X) is the disjoint −1 −1 union of πm (X sing ) and πm (X reg ) = X reg ×X Jm (X), which can be identified with
Jm (X reg ) by Proposition II.6 (because an open immersion is ´etale). The closure of Jm (X reg ) in Jm (X) is therefore a component of the jet scheme Jm (X). One might say this is the “main component” of the jet scheme Jm (X), since it is essentially the component of jets centered on the smooth part of X. This component is thus well understood. For example, Jm (X reg ) is an Am dim X -bundle over X reg and so this component has dimension (m + 1) dim X. −1 On the other hand, the closed set πm (X sing ) may also contribute components to
Jm (X). It does so if and only if it is not contained in the closure of Jm (X reg ). These components, arising from the singularities of X, are the complicated and interesting part of the jet scheme, and they are the main focus of our results. More generally, if X has several components, then each component of X gives rise to a component of Jm (X); that is, the component X0 of X gives rise to the component Jm (X0reg ) of Jm (X). In particular, Jm (X reg ) has precisely the same number of components as X, and Jm (X) has at least as many components as X.
CHAPTER III
Jet Schemes of Monomial Schemes
In this chapter, we analyze the scheme structure of the jet schemes of hypersurface monomial schemes. More specifically, we give a formula for the multiplicity along every component of the jet schemes of a general reduced monomial hypersurface (see Theorem III.1). We also conjecture and prove in several cases a formula for the multiplicity of the irreducible jet schemes of a fat point in the affine line (see Section 3.3). Finally, we give an explicit description of several primary components of the jet schemes of the simple monomial scheme Spec k[x, y]/(xy) (see Proposition III.4 and Theorem III.5). Before we look into these jet schemes, let us say a few words about why monomial schemes are important. First, many computations can be reduced to that of monomial ideals because any ideal can be deformed into a monomial ideal (using Gr¨obner basis theory, see [15, Chapter 15]). Secondly, simple normal crossing divisors are the first class of singularities one encounters when leaving the realm of smooth varieties. Therefore, in trying to understand the structure of jet schemes of singular varieties, we must first understand what happens in this case, where the scheme X is defined by a single square-free monomial.
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16
3.1
Reduced monomial hypersurfaces
We first describe the scheme structure of jet scheme of a reduced monomial hypersurface. In this case, the scheme X ⊆ An is defined by a single monomial x1 · · · xr . Its jet scheme Jm (X) is the closed subscheme of (0)
(m)
(m) Jm (An ) = Spec k[x1 , . . . , x1 , . . . , x(0) n , . . . , xn ],
cut out by polynomials g0 , . . . , gm where gk is the coefficient of tk in the product r Y (0) (1) (m) (xi + xi t + . . . + xi tm ), i=1
see the proof of Proposition II.2. In other words, the ideal (0)
(m)
(m) Jm (X) ⊆ R = k[x1 , . . . , x1 , . . . , x(0) n , . . . , xn ]
has generators of the form gk = where
P
X
(i ) (i )
r) x1 1 x2 2 · · · x(i r
ij = k and 0 ≤ ij ≤ m.
Goward and Smith [17] showed that the minimal primes of Jm (X) are of the form (3.1)
(0)
(t −1)
P (m; t1 , . . . , tr ) = (x1 , . . . , x1 1
where 0 ≤ ti ≤ m + 1 and
P
(tr −1) , . . . , x(0) ) r , . . . , xr
ti = m + 1. (Here, we adopt the convention that the
value ti = 0 means the variable xi does not appear at all.) In particular, Jm (X) is pure dimensional — each component has codimension m + 1. This gives a complete understanding of the irreducible components of the reduced jet schemes of X, but no more information about the scheme structure. The following theorem gives some insight into this scheme structure:
17
Theorem III.1. For X = Spec k[x1 , . . . , xn ]/(x1 · · · xr ), the multiplicity of Jm (X) along P (m; t1 , . . . , tr ) is (m + 1)! . t1 ! · · · tr ! Fix a minimal prime P = P (m; t1 , . . . , tr ) of Jm (X) as described in (3.1) and let R(m; t1 , . . . , tr ) = RP . We want to find `(R(m; t1 , . . . , tr )/Jm (X)). We first make two reductions: (j)
1. Let R0 = k[xi ], 1 ≤ i ≤ n, j ≥ 0, be a polynomial ring in infinitely many indeterminates. Also let R0 (m; t1 , . . . , tr ) = RP0 . Note that the completion of R(m; t1 , . . . , tr ) (respectively R0 (m; t1 , . . . , tr )) at its maximal ideal is isomorphic (0)
(t −1)
to the formal power series ring L[[x1 , . . . , x1 1
(0)
(t −1)
, . . . , x r , . . . , xr r
]], where L
is the residue field of R(m; t1 , . . . , tr ) (respectively R0 (m; t1 , . . . , tr )). Therefore, `(R(m; t1 , . . . , tr )/Jm (X)) = `(R0 (m; t1 , . . . , tr )/Jm (X)). So we may replace R by R0 . 2. Suppose that one of the ti , say tr , is zero — that is, suppose that xr does not appear in P at all. Notice that we can rewrite the generators of Jm (X) as gk =
k X
x(k−q) hq r
with
hq =
X
(i )
(i
r−1 x1 1 · · · xr−1
)
q=0
where
P
ij = q and 0 ≤ ij ≤ m. In other words, g0 = x(0) r h0 (0) g1 = x(1) r h0 + xr h1 (1) (0) g2 = x(2) r h0 + xr h1 + xr h2
etc. (0)
Because xr is a unit in R(m; t1 , . . . , tr−1 , 0), it follows that Jm (X)R(m; t1 , . . . , tr−1 , 0) = (h0 , . . . , hm )R(m; t1 , . . . , tr−1 , 0).
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This means that `(R(m; t1 , . . . , tr−1 , 0)/Jm (X)) = `(R(m; t1 , . . . , tr−1 , 0)/(h0 , . . . , hm )) = `(R(m; t1 , . . . , tr−1 )/Jm (X 0 )) where X 0 = Spec k[x1 , . . . , xn ]/(x1 · · · xr−1 ). So we may assume all ti are positive. To prove Theorem III.1, we also need the following lemma: Lemma III.2. Let R be an arbitrary ring and x1 , . . . , xr nonunits in R. If x1 · · · xr P is a nonzerodivisor on R, then ` (R/(x1 · · · xr )) = ri=1 ` (R/(xi )). Proof. We induce on r. The case r = 1 is trivial. So we may assume that the assertion is true for r − 1. Since x1 · · · xr is a nonzerodivisor on R, each xi is a nonzerodivisor on R. So we have the short exact sequence: 0
/
R/(x1 · · · xr−1 )
·xr
/
R/(x1 · · · xr )
/
/
R/(xr )
Then ` (R/(x1 · · · xr )) = ` (R/(x1 · · · xr−1 )) + `(R/(xr )) =
Pr
i=1
0.
` (R/(xi )) by the
induction hypothesis. Proof of Theorem III.1. We proceed by induction on m. A minimal prime of J0 (X) (0) (j) (0) (0) (j) (0) has the form (xk ) and ` k[xi ](x(0) ) /(x1 · · · xr ) = ` k[xi ](x(0) ) /(xk ) = 1. k
k
This completes the m = 0 case. Now we assume that the assertion is true for m − 1 and consider the multiplicity of Jm (X) along P (m; t1 , . . . , tr ). Note that ht(Jm (X)R(m; t1 , . . . , tr )) = dim(R(m; t1 , . . . , tr )) =
r X
ti
i=1
= m + 1,
(by (3.1))
19
which is the number of generators of Jm (X). So the elements g0 , . . . , gm form a regular sequence on R(m; t1 , . . . , tr ). Now let S = R(m; t1 , . . . , tr )/(g1 , . . . , gm ). Then `(R(m; t1 , . . . , tr )/Jm (X)) = `(S/(g0 )). (0) (0) (0) (0) Since g0 = x1 · · · xr is a nonzerodivisor on S, by Lemma III.2, ` S/(x1 · · · xr ) P (0) (0) = rn=1 ` S/(xn ) . To know what ` S/(xn ) is, we need the following crucial result: (0)
Claim III.3. The ring S/(xn ) is isomorphic to R(m−1; t1 , . . . , tn −1, . . . , tr )/Jm−1 (X). (0)
Proof. Modulo xn , the generators gk (1 ≤ k ≤ m) become gek = where
P
X
(i ) (i )
r) x1 1 x2 2 · · · x(i r
ij = k, 0 ≤ ij ≤ m if j 6= n and 1 ≤ in ≤ m. (q)
(q−1)
By a change of variables, replacing xn by xn
for all q ≥ 1 and fixing the rest,
gek becomes gk−1 . This tells us (0) S/(x(0) f m , xn ) n ) = R(m; t1 , . . . , tr )/(ge1 , . . . , g
∼ = R(m − 1; t1 , . . . , tn − 1, . . . , tr )/(g0 , . . . , gm−1 ) = R(m − 1; t1 , . . . , tn − 1, . . . , tr )/Jm−1 (X). Therefore, by the induction hypothesis, we have `(R(m; t1 , . . . , tr )/Jm (X)) =
r X n=1
m! t1 ! · · · (tn − 1)! · · · tr !
m!(t1 + . . . + tr ) t1 ! · · · tr ! (m + 1)! = . t1 ! · · · tr ! =
20
This gives a better understanding of the jet schemes of the reduced monomial hypersurface. In particular, it gives a feeling for how “fat” the jet schemes are along each of their components. On the other hand, a more precise understanding of the scheme structure in this case seems more difficult. For example, one could hope to describe the generators for each of the primary components. This turns out to be very hard even for the simple monomial scheme X = Spec k[x, y]/(xy).
3.2
Primary components
We now investigate the primary components of the jet schemes Jm (X) when X = Spec k[x, y]/(xy). In this setting, the generators of the defining ideal Jm ⊆ R = k[x0 , . . . , xm , y0 , . . . , ym ] are gk = x0 yk + x1 yk−1 + . . . + xk y0 for k ranging from 0 to m, and the minimal primes of Jm are P = (x0 , . . . , xc−1 , y0 , . . . , ym−c ) where 0 ≤ c ≤ m + 1. First, we deal with the easiest components. Proposition III.4. With notation as above, the scheme Jm (X) is reduced along the components defined by the prime ideals P = (x0 , . . . , xm ) and P = (y0 , . . . , ym ). That is, the P -primary component of Jm (X) in this case is P . Proof. Let P = (x0 , . . . , xm ). Localizing at P , we should regard every yi as a unit, and the generators gk as linear polynomials in the xi ’s. They are obviously independent, and generate P RP . This shows that the P -primary ideal in the decomposition of Jm is P itself. Alternatively, note that by Theorem III.1, the multiplicity of the
21
scheme Jm (X) along this component is 1. In this way, we can also conclude that Jm RP ∩ R = P . It is also instructive to interpret Proposition III.4 geometrically. Recall from Section 2.4 that we have a natural projection πm : Jm (X) → X, which allows us to decompose Jm (X) into the sets lying over the smooth part of X and over the singular part. If we consider the open set U = X \ V(x) (the y-axis with the origin removed), −1 (U ) in Jm (X) is the component of Jm (X) defined then we see that the closure of πm
by P = (x0 , . . . , xm ). So this component of Jm (X) naturally corresponds to the jet scheme of the y-axis, while by symmetry, the component defined by P = (y0 , . . . , ym ) corresponds to the jet scheme of the x-axis. Each of these components is reduced and irreducible. But all other components lie over the singular locus (the origin), and have more complicated scheme structure associated with them. By Theorem III.1, the mul tiplicity along the component defined by (x0 , . . . , xc−1 , y0 , . . . , ym−c ) is m+1 , but c beyond that they are quite complicated. The next theorem gives explicit equations for one of them. Theorem III.5. With notation as in the first paragraph of Section 3.2, consider P = (x0 , y0 , . . . , ym−1 ), a particular minimal prime of Jm . Let Q be the ideal Q = (g0 , . . . , gm , xm+1 , 0 yi yj : 0 ≤ i ≤ j ≤ m − 1 and i + j ≤ m − 1, yi yj − yi+j−m ym : 1 ≤ i ≤ j ≤ m − 1 and i + j ≥ m) Then Q is the primary ideal corresponding to the minimal prime P in the primary decomposition of Jm .
22
Since P is a minimal prime of Jm , its P -primary component is precisely Jm RP ∩R. Our goal is therefore to show that Q = Jm RP ∩ R. We will achieve this in two steps: 1. Jm RP = QRP 2. Q is P -primary (and therefore Q = QRP ∩ R) The key to both steps is to show that Jm RW = QRW for some suitable multiplicative system W disjoint from P . To find W , we first find a Gr¨obner basis for Q and so get a vector space basis for R/Q. Then using this basis, we obtain some nonzerodivisors on R/Q which will be used to generate W . Before we dive into the proof of Theorem III.5, let us briefly review some Gr¨obner basis theory. A more detailed discussion can be found in Chapter 15 of [15]. We start with some definitions necessary to define a Gr¨obner basis. Let S = k[z1 , . . . , zn ] be a polynomial ring. Definition III.6. A monomial order < on S is a total order on the set of all monomials in S satisfying the property: If u < v and w 6= 1 is any monomial, then u < uw < vw. Definition III.7. Given a monomial order < on S, then for any f ∈ S, we define the initial term of f , denoted by in(f ), to be the greatest term of f with respect to the order xm−1 > · · · > x0 > ym > · · · > y0 under graded reverse lexicographic order. Graded reverse lexicographic order is a monomial order used by computer algebra packages because it often makes Gr¨obner basis calculations more efficient [3, §4].
24
Definition III.12. Let S = k[z1 , . . . , zn ] be a polynomial ring and α = z1a1 · · · znan , β = z1b1 · · · znbn be any two monomials in S. The graded reverse lexicographic order on the monomials of S with z1 > z2 > . . . > zn is defined by α < β if and only if deg α < deg β or deg α = deg β and ai > bi for the last index i with ai 6= bi . Example III.13. Let us compute the initial term of the generators of Q with respect to the orders given in Proposition III.11. First, let us introduce some notation. We write for 0 ≤ i, j ≤ m − 1 and i + j ≤ m − 1
(3.2)
Fij = yi yj
(3.3)
Gij = yi yj − yi+j−m ym
for 1 ≤ i, j ≤ m − 1 and i + j ≥ m.
Now the initial terms of the generators of Q are (3.4)
in(gk ) = x0 yk
(3.5)
in(Fij ) = yi yj
(3.6)
in(Gij ) = yi yj
since yk > yl for all l < k
since i, j < m implies that yi+j−m < yi , yj .
Proof of Proposition III.11. By Buchberger’s criterion, it suffices to show that for each pair of generators fi , fj with gcd(in(fi ), in(fj )) 6= 1, we have an expression X ij in(fj ) in(fi ) fi − fj = hk fk gcd(in(fi ), in(fj )) gcd(in(fi ), in(fj )) k in(fj ) ij where hij k ∈ R and in(hk fk ) < in gcd(in(fi ),in(fj )) fi for all k. (3.7)
), in(Gij )) are First of all, notice that both gcd(in(xm+1 ), in(Fij )) and gcd(in(xm+1 0 0 equal to 1, for any i and j. So we only have to check Buchberger’s Criterion for the remaining pairs of generators of Q. Buchberger’s Criterion for pairs (gi , gj ): Without loss of generality, we may assume 0 ≤ i < j ≤ m. Then plugging (gi , gj ) into the left hand side of (3.7), we
25
have x0 yj x0 y i gi − gj x0 x0 j i X X = xk yi−k yj − xl yj−l yi k=1
l=1
X
=
xk (yi−k yj − yj−k yi ) +
1≤k≤i i+j−k≤m−1
X
=
j X
xl (yi−l yj − yj−l yi ) −
1≤l≤i i+j−l≥m
(xk Fi−k,j − xk Fj−k,i ) +
1≤k≤i i+j−k≤m−1
−
X
X
j X
xq yj−q yi
q=i+1
(xl Gi−l,j − xl Gj−l,i ) −
i X
1≤l≤i i+j−l≥m j≤m−1
xl Gm−l,i
l=1
xq Fj−q,i
q=i+1
We can easily see that in(xk Fi−k,j ) = xk yi−k yj
and
in(xk Fj−k,i ) = xk yj−k yi ,
in(xl Gi−l,j ) = xl yi−l yj
and
in(xl Gj−l,i ) = xl yj−l yi ,
in(xl Gm−l,i ) = xl ym−l yi
and
in(xq Fj−q,i ) = xq yj−q yi
are all less than in(yj gi ) = x0 yi yj . Buchberger’s Criterion for pairs (gi , xm+1 ): Plugging (g0 , xm+1 ) into the left hand 0 0 m+1 side of (3.7), we get 0, which is definitely less than in(xm y0 . So we may 0 g 0 ) = x0
assume 1 ≤ i ≤ m. Then substituting the pair (gi , xm+1 ) into the left hand side of 0 (3.7), we have (3.8)
x0 yi m+1 xm+1 0 m m gi − x = x1 xm 0 yi−1 + x2 x0 yi−2 + . . . + xi x0 y0 . x0 x0 0
Thus it suffices to show that each of these terms can be expressed in terms of the generators of Q with initial terms less than xm+1 yi . To achieve this, we use: 0
26
Claim III.14. For 0 ≤ n < i and all k ≥ 1, xi1 · · · xik xn+1 yn = 0
P
hjn gj where
in(hjn gj ) < xk+n+1 yi . 0 Proof. We induce on n. When n = 0, xi1 · · · xik x0 y0 = xi1 · · · xik g0 , and clearly xi1 · · · xik x0 y0 < xk+1 0 yi . Now assume n ≥ 1. Then yn = xi1 · · · xik xn0 gn xi1 · · · xik xn+1 0 n−1 X
−
i
−1
n−ik+1 +1
x0k+1 (xi1 · · · xik xik+1 x0
yn−ik+1 )
ik+1 =1
− xi1 · · · xik xn−1 xn g0 . 0 We just need to check that each of these terms satisfies the condition stated in our claim. Notice that in(xi1 · · · xik xn0 gn ) = xi1 · · · xik xn+1 yn 0 in(xi1 · · · xik x0n−1 xn g0 ) = xi1 · · · xik xn0 xn y0 are both less than xk+n+1 yi . Moreover, by the induction hypothesis, we can write 0 P n−i +1 k+1+n−ik+1 +1 xi1 · · · xik xik+1 x0 k+1 yn−ik+1 = hj,n−ik+1 gj where in(hj,n−ik+1 gj ) < x0 yi . i
−1
i
Since in(x0k+1 hj,n−ik+1 gj ) = x0k+1
−1
in(hj,n−ik+1 gj ), we have proven the claim.
Now we are in a position to verify that the expression (3.8) can be expressed in the desired from as described in the right hand side of (3.7). Claim III.14 tells us we can express xk x0i−k+1 yi−k =
X
hik j gj
i−k+2 yi for 1 ≤ k ≤ i. Thus, we can write where in(hik j gj ) < x0 m−i+k−1 xk xm 0 yi−k = x0
X
hik j gj
m−i+k−1 i−k+2 with in(xm−i+k−1 hik (x0 yi ) = xm+1 yi = in(xm 0 0 j gj ) < x0 0 gi ). Therefore, the
pairs (gi , xm+1 ) satisfy Buchberger’s Criterion. 0
27
Buchberger’s Criterion for pairs (gk , Fij ): From (3.4) and (3.5), we see that gcd(in(gk ), in(Fij )) = 1 if k = m or k 6= i, j. So without loss of generality, we may assume gk = gi for 0 ≤ i ≤ m − 1, in which case gcd(gi , Fij ) = yi . Now plug (gi , Fij ) into the left hand side of (3.7) to obtain yi yj x 0 yi gi − Fij = x1 Fi−1,j + x2 Fi−2,j + . . . + xi F0j . yi yi Clearly, in(xq Fi−q,j ) = xq yi−q yj < in(yj gi ) = x0 yi yj for 1 ≤ q ≤ i. Buchberger’s Criterion for pairs (gk , Gij ): Referring to (3.4) and (3.6), we see that gcd(in(gk ), in(Gij )) = 1 when k 6= i, j, we may assume gk = gi for 1 ≤ i ≤ m − 1. Then plugging (gi , Gij ) into the left hand side of (3.7) gives x0 y i yi yj gi − Gij yi yi = x1 yi−1 yj + x2 yi−2 yj + . . . + xi y0 yj + x0 yi+j−m ym m−j−1
=
X
xi−l yl yj + ym gi+j−m +
m−j−1
X
xi−q (yq yj − yq+j−m ym )
q=m−j
l=0
=
i−1 X
xi−l Flj + ym gi+j−m +
i−1 X
xi−q Gqj
q=m−j
l=0
Here, we can see that in(xi−l Flj ) = xi−l yl yj , in(ym gi+j−m ) = x0 ym yi+j−m , in(xi−q Gqj ) = xi−q yq yj are all less than in(yj gi ) = x0 yi yj . Buchberger’s Criterion for pairs (Fij , Glk ): We have gcd(in(Fij ), in(Glk )) = 1 if {i, j} ∩ {l, k} = ∅. So we may assume l = i. By substituting the pair (Fij , Gik ) into the left hand side of (3.7), we get yi yj yi yk Fij − Gik = ym Fj,i+k−m yi yi
28
and in(ym Fj,i+k−m ) = ym yj yi+k−m < in(yk Fij ) = yk yi yj . We have checked that Buchberger’s criterion is satisfied by all the generators of Q described in Theorem III.5. Thus we may conclude that they form a Gr¨obner basis for Q. Now we are ready to begin the proof of Theorem III.5. As outlined immediately after the theorem, we first need to show that Jm RP = QRP . The following lemma is required. Lemma III.15. The elements x1 , . . . , xm , ym are nonzerodivisors on R/Q. Proof. The idea is that if xi (respectively ym ) is a zerodivisor on R/Q, then there exists f 6= 0 such that xi f = 0 (respectively ym f = 0) in R/Q. We will write f and xi f (respectively ym f ) in terms of a “suitable basis” for R/Q and arrive at a contradiction. Since we already have a Gr¨obner basis for Q, there is a natural vector space basis for R/Q, namely the set of all monomials not in the initial ideal of Q [15, Theorem 15.3]. By Proposition III.11 and Example III.13, the initial ideal of Q is (3.9)
in(Q) = (x0 y0 , . . . , x0 ym , xm+1 , yi yj : 0 ≤ i ≤ j ≤ m − 1). 0
jm A monomial α = xi00 · · · ximm y0j0 · · · ym is in the ideal α ∈ in(Q) if and only if α is
divisible by one (or more) of the minimal generators for in(Q) listed in (3.9). Thus, the monomials not in in(Q) are of the form (3.10)
xj0 xi11 · · · ximm
or
k xi11 · · · ximm yj ym
where 0 ≤ j ≤ m, il , k ≥ 0 and = 0 or 1. These monomials form a convenient basis for analyzing the ring R/Q.
29
We first show that x1 , . . . , xm are nonzerodivisors on R/Q. Suppose xi f = 0 in R/Q. Since Q is a homogeneous ideal, we may assume f is homogeneous and write f in terms of the basis elements in (3.10). Note that the set of monomials listed in (3.10) is stable under multiplication by xi . So xi f = 0 implies f = 0 in R/Q. Thus, x1 , . . . , xm are nonzerodivisors on R/Q. We now show that ym is a nonzerodivisor on R/Q. Suppose on the contrary that ym f = 0 in R/Q. It is convenient to consider the bi-grading on the polynomial ring k[x0 , . . . , xm , y0 , . . . , ym ], assigning weight (1, 0) to xi and (0, 1) to yi . Since the generators for Q are homogeneous with respect to this N2 -grading, there is no loss of generality in assuming f is homogeneous of bi-degree (d1 , d2 ). Note that i0
i0
k deg(xj0 xi11 · · · ximm ) = (j + i1 + . . . + im , 0) 6= (i01 + . . . + i0m , + k) = deg(x11 · · · xmm yj ym ).
So, since f can be assumed bihomogeneous, we have two cases: i0
i0
k Case 1: The element f involves only basis elements of the form x11 · · · xmm yj ym in
(3.10). In this case, all the monomials in our expression for ym f are still basis elements of the form listed in (3.10). So ym f = 0 yields f = 0 in R/Q, which means ym does not annihilate elements f of this type. Case 2: The element f involves only basis elements of the form xj0 xi11 · · · ximm from (3.10). Suppose f =
m X
X
j=0 j+
λji xj0 xi11 · · · ximm 6= 0 in R/Q. By the orderings described
Pi i=d1
in Proposition III.11, the initial term of ym f must be of the form xj0 xi11 · · · ximm ym with j minimal. Because ym f ∈ Q, we have in(ym f ) ∈ in(Q), which implies all λ0i are zero (see (3.9) for a set of generators of in(Q)). To get a contradiction, we will write ym f as a linear combination of the basis elements of R/Q and show that it is
30
nonzero. For this, we need the following: Claim III.16. For 1 ≤ j ≤ m, j + 1 ≤ k ≤ m and all il ≥ 0, we can express xj0 xi11 · · · ximm yk = (−1)j xi11 +j xi22 · · · ximm yk−j + β (mod Q), where each term of β involves some of x1 , . . . , xm and one of y0 , . . . , yk−j−1 . Proof. Since Jm ⊂ Q, it is enough to show the congruence modulo Jm . We do this by induction on k. For k = 1, from the definition of gk , we have x0 xi11 · · · ximm yk = xi11 · · · ximm (x0 gk − x1 yk−1 − x2 yk−2 − . . . − xk y0 ). So modulo Jm , we can express x0 xi11 · · · ximm yk = −xi11 +1 xi22 · · · ximm yk−1 + terms involving some of x1 , . . . , xm and one of y0 , . . . , yk−2 . For the inductive step, we have i1 im xj0 xi11 · · · ximm yk = x0 (xj−1 0 x1 · · · xm y k )
= x0 [(−1)j−1 xi11 +j−1 xi22 · · · ximm yk−j+1 + β1 ] (mod Jm ) where each term of β1 involves some of x1 , . . . , xm and one of y0 , . . . , yk−j . Notice that we can apply the induction hypothesis to every term of x0 β1 , and therefore each term of x0 β1 involves some of x1 , . . . , xm and one of y0 , . . . , yk−j−1 . Now by the definition of gk−j+1 , we can rewrite (−1)j−1 x0 xi11 +j−1 xi22 · · · ximm yk−j+1 = (−1)j−1 xi11 +j−1 xi22 · · · ximm (gk−j+1 − x1 yk−j − . . . − xk−j+1 y0 ). So by induction, we can express xj0 xi11 · · · ximm yk = (−1)j xi11 +j xi22 · · · ximm yk−j + β (mod Jm ), where each term of β involves some of x1 , . . . , xm and one of y0 , . . . , yk−j−1 . This finishes the proof of our claim.
31
Now let N be the largest integer such that λN i 6= 0 for some i. Then Claim III.16 tells us we can express ym f as X N+
Pi
λN i xi11 +N · · · ximm ym−N + linear combination of other basis elements of R/Q
i=d1
which is nonzero in R/Q, contradiction. So ym also does not annihilate elements f of this type. Combining the results in both cases, we show that ym is a nonzerodivisor on R/Q, as desired. Lemma III.17. With notation as in Theorem III.5, the ideals Jm RP and QRP are the same. Proof. It suffices to show Jm RW = QRW for any multiplicative system W disjoint from P , since the equality is preserved under further localization. Let W be the multiplicative system generated by x1 , . . . , xm , ym . By Lemma III.15, x1 , . . . , xm , ym are nonzerodivisors on R/Q. Therefore, we have an inclusion R ,→ RW . Since Jm ⊆ Q, obviously Jm RW ⊆ QRW . So we only need to show the reverse inclusion. We do this in three steps, according to the different types of generators of Q (notation as in Theorem III.5). Step 1: The element xm+1 is in Jm RW . 0 Proof. We show by induction on i that xi+1 0 yi ∈ Jm for 0 ≤ i ≤ m. For i = 0, clearly x0 y0 ∈ Jm , so there is nothing to show. For i ≥ 1, note that i i−1 i−2 2 i−1 xi0 gi = xi+1 0 yi + x1 (x0 yi−1 ) + x0 x2 (x0 yi−2 ) + . . . + x0 xi−1 (x0 y1 ) + x0 xi g0 .
By the induction hypothesis, all terms in parenthesis are in Jm . Thus, xi+1 0 yi ∈ Jm , completing the induction step. Taking i to be m, we have xm+1 ym ∈ Jm . Since we have inverted ym in RW , we 0 conclude that xm+1 ∈ Jm RW . 0
32
Step 2: The elements yi yj are in Jm RW for all 0 ≤ i, j ≤ m − 1 and i + j ≤ m − 1. Proof. We proceed by induction on i+j, and for fixed i+j, subinduction on i. When i + j = 0, i = j = 0. Then x1 y02 = y0 g1 − y1 g0 ∈ Jm . Since x1 is a unit in RW , y02 ∈ Jm RW . This completes the base case i + j = 0. For general i and j, consider the polynomial yi gj+1 − yj+1 gi ∈ Jm RW . In this difference, the terms x0 yi yj+1 cancel. After regrouping, the polynomial ! j+1 i X X xk yj+1−k yi − xl yi−l yj+1 . yi gj+1 − yj+1 gi = x1 yi yj − x1 yi−1 yj+1 + k=2
l=2
Observe that the term x1 yi−1 yj+1 is in Jm RW , by induction on i. Moreover, all the terms in the parenthesis involve products yi0 yj 0 with i0 + j 0 < i + j , and so are in Jm RW by induction. Since x1 is a unit in RW , the element yi yj is in Jm RW . Step 3: The elements {yi yj − yi+j−m ym : 1 ≤ i, j ≤ m − 1, i + j ≥ m} are in Jm RW . Proof. It suffices to show that yi yj − yk yl ∈ Jm Rw whenever i + j = k + l ≥ m. Without loss of generality, we assume i ≤ j. We induce on i + j. Consider again the polynomial yi gj+1 − yj+1 gi ∈ Jm RW . Notice that the x0 yi yj+1 terms cancel again, and this expression can be written as (3.11)
x1 (yi yj − yi−1 yj+1 ) +
i X k=2
xk (yj+1−k yi − yi−k yj+1 ) +
j+1 X
xl yj+1−l yi .
l=i+1
Since k ≥ 2, the coefficients of xk in expression (3.11) are all in Jm RW , by induction on i + j (or by Step 2 if i + j + 1 − k < m). Furthermore, l ≥ i + 1 implies that
33
i + j + 1 − l ≤ j ≤ m − 1, and so all terms in the last summation of (3.11) are in Jm RW by Step 2. Since x1 is a unit in RW , we conclude that also yi yj − yi−1 yj+1 is in Jm RW . By adding polynomials of this type, we infer that all binomials of the form yi yj − yk yl with i + j = k + l ≥ m are in Jm RW . Putting these three steps together, we see that all the generators of QRW are in Jm RW . Thus, Jm RW = QRW . Since W is disjoint from P , we have shown that Jm RP = QRP . Continuing with the outline to prove Theorem III.5, we now need the following lemma: Lemma III.18. With P and Q as defined in Theorem III.5, Q is P -primary. Proof. By definition, we must show that P is the radical of Q and also R/Q is √ coprimary, that is, that all nonzerodivisors are nilpotent. To see that Q = P , note √ that since Q ⊆ P , it suffices to show P ⊆ Q. First, notice that xm+1 ∈ Q gives 0 √ √ x0 ∈ Q. We induce on i to show that yi ∈ Q for 0 ≤ i ≤ m − 1. Obviously, √ y02 ∈ Q implies y0 ∈ Q. Now for i ≥ 1, we have yi−1 yi+1 − yi2 ∈ Q. So by √ √ induction, assuming yi−1 ∈ Q, then it follows that also yi ∈ Q. This shows that √ √ P = (x0 , y0 , . . . , ym−1 ) ⊆ Q and hence P = Q. It remains to show that R/Q is coprimary. The following lemma is helpful. Lemma III.19. For any ring S and ideal I of S, S/I is coprimary if and only if (S/I)x is coprimary for any nonzerodivisor x on S/I. Proof. Suppose S/I is coprimary and say
r xn
· xsm = 0 in (S/I)x . Then there exists an
integer N such that xN rs = 0 in S/I. Since x is a nonzerodivisor on S/I, we have rs = 0 in S/I, which implies r = 0 or sa = 0 in S/I for some a > 0. So s a = 0 in (S/I)x , showing (S/I)x is coprimary. m x
r xn
= 0 or
34
For the other implication, suppose rs = 0 in S/I. Then 1r · 1s = 0 in (S/I)x implies r 1
= 0 or
sa 1
= 0 in (S/I)x for some a > 0. So there exists an integer N such that
xN r = 0 or xN sa = 0 in S/I. Thus, x nonzerodivisor on S/I gives us r = 0 or sa = 0 in S/I. This completes the proof of Lemma III.19. In our case, Lemma III.19 says showing R/Q is coprimary is equivalent to showing (R/Q)W is coprimary. But we have proven earlier that Jm RW = QRW , which allows to reduce to checking RW /Jm RW is coprimary. The idea is to simplify the presentation of the ring RW /Jm RW to a ring which we can easily recognize to be coprimary. To do this, we need the following claim: −1 −1 Claim III.20. Let T = k[x1 , x−1 1 , . . . , xm , xm , ym , ym ]. Given λj ∈ T with λ1 a
unit, we have an isomorphism T [x0 , y0 , . . . , ym−i ] (g0 , . . . , gm−i , ym−i − λ1 xi0 − λ2 ym−i−1 − . . . − λm−i+1 y0 ) T [x0 , y0 , . . . , ym−i−1 ] ∼ = (g0 , . . . , gm−i−1 , ym−i−1 − µ1 xi+1 − µ2 ym−i−2 − . . . − µm−i y0 ) 0 for some µj ∈ T and µ1 a unit, for i ranging from 1 to m − 1. Proof. First note that ym−i only appears in gm−i but not the gk for k < m − i. Now plugging ym−i = λ1 xi0 + λ2 ym−i−1 + ... + λm−i+1 y0 into gm−i , we get gm−i = (λ1 xi+1 0 + λ2 x0 ym−i−1 + ... + λm−i+1 x0 y0 ) + x1 ym−i−1 + x2 ym−i−2 + . . . + xm−1 y0 . But using gj = 0, we can replace x0 yj by −x1 yj−1 − . . . − xj y0 for every 0 ≤ j ≤ m − i − 1. After simplification, we have gm−i = x1 ym−i−1 + λ1 xi+1 + λ2 0 ym−i−2 + . . . + λ0m−i y0 0 where λ0j ∈ T and λ1 is a unit in T . Now taking µ1 =
λ1 x1
and µj =
λ0j x1
(note x1 is a
unit in T ) yields the desired result. This completes the proof of the claim.
35
Continuing with the proof of Lemma III.18, we seek to show RW /Jm RW is coprimary. By repeating applications of Claim III.20, we have RW T [x0 , y0 , · · · , ym−1 ] = Jm RW (g0 , . . . , gm−1 , ym−1 − λ1 x0 − λ2 ym−2 − . . . − λm y0 ) T [x0 , y0 ] ∼ = (g0 , y0 − µxm 0 ) T [x0 ] ∼ = m+1 (x0 ) Finally, T being a localization of the domain k[x1 , . . . , xm , ym ] implies that T is a domain and so we can see easily that T [x0 ]/(xm+1 ) is coprimary. Hence, (R/Q)W is 0 coprimary. To summarize, we have now proved Theorem III.5. Indeed, Lemma III.18 tells us Q = QRP ∩ R. Combining this with Lemma III.17, we have Q = QRP ∩ R = Jm RP ∩ R. In other words, the ideal Q is the primary ideal corresponding to the minimal prime P of Jm , as asserted.
3.3
Fat points on the affine line
Another case I have studied extensively, inspired by a question of Ein, is the case of a fat point in the affine line. In this case, X = Spec k[x]/(xn ) for some n ≥ 2. Because an m-jet of X is a k-morphism Spec k[t]/(tm+1 ) → Spec k[x]/(xn ), it corresponds to a map of k-algebras k[x]/(xn ) → k[t]/(tm+1 ) (3.12)
x 7→ xa ta + xa+1 ta+1 + . . . + xm tm where xa 6= 0,
36
subject to the condition that (xa ta + xa+1 ta+1 + . . . + xm tm )n = tna (xa + xa+1 t + . . . + xm tm−a )n = 0 (mod tm+1 ). Therefore, a particular choice of xa , . . . , xm defines an m-jet if and only if na ≥ m+1. So as a set, we can identify Jm (X) with the closed subset of Spec k[x0 , . . . , xm ] = Jm (A1 ) defined by x0 , . . . , xb mn c , where b m c is the greatest integer less than or equal n to m/n. In particular, the jet scheme Jm (X) is irreducible; that is, the ideal Jm (X) ⊆ R = k[x0 , . . . , xm ] has a unique minimal prime (3.13)
P = (x0 , . . . , xb mn c ).
However, the scheme structure of Jm (X) is much more complicated. From (3.12), we see that the jet scheme Jm (X) is defined by the polynomials gk ∈ R that are the coefficients of tk in the product (x0 + x1 t + . . . + xm tm )n . In particular, we have a0 + a1 + . . . + am = n X a0 a1 am (3.14) gk = ca x 0 x 1 · · · x m where a1 + 2a2 + . . . + mam = k n are multinomial coefficients. and ca = a0 ,...,a m Ein asked whether we could give a formula for the multiplicity of Jm (X) along its unique irreducible component as a function of m and n. In this section, we propose and prove several cases of the following conjecture. Conjecture III.21. For X = Spec k[x]/(xn ), the multiplicity of Jm (X) is n + r − m + nr r+1 where r = b m c, the largest integer less than or equal to n
m . n
The first few values of the multiplicity generated by Macaulay 2 are illustrated in the chart below.
37
J0 (X) J1 (X) J2 (X)
J3 (X)
J4 (X) J5 (X) J6 (X) J7 (X)
J8 (X)
n=2
2
1
3
1
4
1
5
1
6
n=3
3
2
1
6
3
1
10
4
1
n=4
4
3
2
1
10
6
3
1
20
3.3.1
Proof of conjecture when m < n
In this section, we will prove Conjecture III.21 in the case when m < n. This is the easiest case because b m c = 0 and so the radical of Jm (X) is simply (x0 ). The n argument boils down to explicitly calculating the highest power of x0 which divides each generator. Theorem III.22. Let X be the scheme Spec k[x]/(xn ). When m < n, the multiplicity of Jm (X) is n − m, as predicted by Conjecture III.21. Proof. First, notice that
p Jm (X) = (x0 ). Next, we claim that x0n−k divides gk for
all k, but no higher power of x0 does. To see this, consider a monomial term of gk as in (3.14). We will first show that a0 ≥ n − k, and then that one of the monomial terms of gk is xn−k · u where u is a unit in R(x0 ) . 0 Suppose a0 < n − k. Then the condition a0 + a1 + . . . + am = n implies that a1 + . . . + am > k = a1 + 2a2 + . . . + mam . This is impossible because all ai ≥ 0. So a0 ≥ n−k. Now if a0 = n−k, then the two conditions on ai ’s force a2 = · · · = am = 0 and a1 = k. Therefore, gk has the monomial term x0n−k xk1 and x1 is a unit in R(x0 ) . So our claim yields , . . . , xn−m ) R(x0 ) /Jm (X) = R(x0 ) /(xn0 , xn−1 0 0 = R(x0 ) /(x0n−m )
38
Thus, the multiplicity of Jm (X) is n − m. 3.3.2
Proof of conjecture in the case when m ≡ −1 mod n
Now we turn our attention to the case when m ≡ −1 mod n, showing that the multiplicity in this case is one, as conjectured. We will demonstrate this by showing that a subset of the generators of Jm (X) generate the maximal ideal P RP in the local ring RP . Theorem III.23. With notation as in Conjecture III.21, the multiplicity of Jm (X) is 1 when m ≡ −1 mod n. Proof. Write m = nq − 1 for some q. By Theorem III.22, we may assume that q ≥ 2. p c = q − 1, Jm (X) = P = (x0 , . . . , xq−1 ) (see the discussion preceding Since b m n (3.13)). To show that Jm (X) is reduced, we focus our attention on the last q generators g(n−1)q , . . . , gnq−1 and show that they form a regular system of parameters in RP . Indeed, we will show that the images of these elements in P RP /P 2 RP form a basis for the cotangent space P/P 2 of the regular local ring RP . From the expression (3.14), we easily check that the images of g(n−1)q , . . . , gnq−1 in P/P 2 take the form g(n−1)q = n(xq )n−1 x0 g(n−1)q+1 = λ(n−1)q+1,0 x0 + n(xq )n−1 x1 .. .
.. .
gnq−1 = λnq−1,0 x0 + . . . + λnq−1,q−2 xq−2 + n(xq )n−1 xq−1 where the λij are elements of the field RP /P RP and we use xi to denote the image of xi modulo P 2 . Because (xq )n−1 is a unit in RP /P RP , it is easy to see that the elements g(n−1)q , . . . , gnq−1 form a basis for the RP /P RP -vector space P/P 2 .
39
In other words, we have a basis for the cotangent space P/P 2 , or equivalently, g(n−1)q , . . . , gnq−1 form a regular system of parameters in RP . Thus, Jm (X)RP ⊇ (g(n−1)q , . . . , gnq−1 )RP = P RP yields `(RP /Jm (X)) = 1.
3.3.3
A lower bound in the case when m ≡ −2 mod n
In this section, we give a lower bound for the multiplicity of the scheme Jm (X) when m ≡ −2 mod n. Theorem III.24. Let X = Spec k[x]/(xn ). The multiplicity of Jm (X) along its unique component is at least b m c + 2. n Before we prove this theorem, we first need to establish the following fact. Lemma III.25. Let R = L[[x0 , . . . , xq−1 ]] be a formal power series ring over a field L of characteristic zero. Consider any L-algebra map φ : R → L[[y]] xi 7→ fi (y) such that fq−1 (y) = y. Then for any g = g(x0 , . . . , xq−1 ) ∈ R, the N th derivative of image of g under φ with respect to y is N ∂kg dak fik dN φ(g) X da1 fi1 da2 fi2 = · · · · · · · dy N ∂xi1 · · · ∂xik (f0 (y),...,fq−1 (y)) dy a1 dy a2 dy ak k=1 P where 0 ≤ ij ≤ q − 1, aj ≥ 1 and aj = N . In particular, N ak X da1 fi1 ∂kg d f dN φ(g) i k · . (3.15) = · · · · · dy N y=0 k=1 ∂xi1 · · · ∂xik 0 dy a1 y=0 dy ak y=0 Proof. To simplify notation, we write < f (y) > for (f0 (y), . . . , fq−1 (y)). We prove by induction on N . Clearly, q−1 dφ(g) X ∂g dfi = · . dy ∂x dy i i=0
40
Now for the inductive step, since ! q−1 k+1 k X ∂ g d ∂ g dfj = · dy ∂xi1 · · · ∂xik ∂xi1 · · · ∂xik ∂xj dy j=0 and d dy
daj fij dy aj
daj +1 fij = , dy aj +1
we see that the N th derivative of φ(g) with respect to y has the desired form. Proof of Theorem III.24. Let q be such that m = nq −2. We may assume that q ≥ 2, p by Theorem III.22. Note that Jm (X) = P = (x0 , . . . , xq−1 ), same as the previous case. Since RP /Jm (X) is a 0-dimensional ring, it is isomorphic to its completion at its maximal ideal, so we may replace RP with its completion L[[x0 , . . . , xq−1 ]] where L is isomorphic to the residue field RP /P RP . We will compute `(RP /Jm (X)) in two steps: 1. Using the relations from the last q − 1 generators of Jm (X), we can express x0 , . . . , xq−2 in terms of xq−1 modulo (g(n−1)q , . . . , gnq−2 ). 2. Rewriting the remaining generators of Jm (X), we can factor out xq+1 q−1 from each of them. Since the minimal prime P is the same as in the previous case, the images of g(n−1)q , g(n−1)q+1 , . . . , gnq−2 in the cotangent space P/P 2 are the same as listed in the proof of Theorem III.23. Therefore, we can see that g(n−1)q , . . . , gnq−2 , xq−1 form a regular system of parameters in RP . So the quotient S = RP /(g(n−1)q+k : 0 ≤ k ≤ q − 2) is a regular local ring with uniformizing parameter xq−1 . This means that we can write xi = µ(xq−1 )Ni for some unit µ in S and Ni ≥ 1 for each i = 0, . . . , q − 2. To find the multiplicity, we need to learn about Ni .
41
Claim III.26. For 0 ≤ i ≤ q − 2, Ni ≥ q − i. Proof. For a fixed i, to show Ni ≥ q − i is equivalent to showing that
dN xi dxq−1 N 0
=0
for all 0 ≤ N ≤ q − i − 1. (Here, the origin is the point given by the minimal prime P .) For the purpose of differentiation, we write y for xq−1 and consider the L-algebra map φ : RP = L[[x0 , . . . , xq−1 ]] → L[[y]] = S xi 7→ fi (y) such that fq−1 (y) = y and φ(g(n−1)q ) = · · · = φ(gnq−2 ) = 0. Then our goal is to N = 0 for all 0 ≤ N ≤ q − i − 1. Note that the case N = 0 states show that ddyNfi y=0
fi (0) = 0, which is obvious. We proceed by double induction on N and i, assuming for each pair (N, i), that the statement holds for all (nonnegative) (N 0 , i0 ) with N 0 < N or N 0 = N and i0 < i. Since φ(gk ) is zero for (n − 1)q ≤ k ≤ nq − 2, its N th derivative
dN φ(gk ) dy N
is also zero.
We will carefully select one such gk for which we also assume (by induction) that all but one of the terms in the right hand side of (3.15) vanish. The consequent dN fi vanishing of the remaining term will force dyN . The right choice of gk in each y=0
case turns out to be gri , where ri = (n − 1)q + i. dN φ(gri ) Fix N and i, and consider dyN as described in the lemma. In particular, y=0
we consider the term ak da1 fi1 ∂ k gri d f i k · . · · · · · ∂xi1 · · · ∂xik 0 dy a1 y=0 dy ak y=0 daj fi If this term is nonzero, then each dyajj is nonzero. By the induction hypothy=0
esis (which doesn’t apply if aj = N , but see below), this means that aj ≥ q − ij . k k X X That is, ij ≥ q − aj . Summing yields ij ≥ (q − aj ), but the condition on aj j=1
j=1
42
from Lemma III.25 simplifies this to
P
ij ≥ kq − N . We haven’t yet used our choice
of ri . Note that
∂ k gri ∂xi1 ···∂xik
0
6= 0 if and only if gri has a term α = λxi1 · · · xik with λ ∈ / P.
The λ with the smallest degree would have to be xqn−k , but even this term has degree at least X
ij + q(n − k) ≥ kq − N + q(n − k) = qn − N > ri ,
because N ≤ q − i − 1 and ri = nq − q + i. This is too high to appear in gri , so the derivative term is zero. Note that in the neglected a1 = N case, either i1 < i, which gives us zero derivative by induction, or i1 > i, which gives us the above inequality i1 + q(n − k) > nq − q + i = ri leading to zero derivative, or finally i1 = i. Thus the ∂gri d N fi only remaining derivative term is ∂xi · dyN , which must therefore also be zero. 0 y=0 ∂gri dN fi n−1 = 0, as desired. But ∂xi = xq 6= 0. So dyN 0
y=0
As a result, we have xi = xq−1 q−i hi (xq−1 ) for i = 0, . . . , q − 2. Note that hi (xq−1 ) can be a unit in S. Now the gk ’s remaining in S are exactly those with k ≤ n(q − 1) − 1. We need to show that each of these is divisible by the q + 1 power of the uniformizing parameter xq−1 . Recall the generator gk in RP has the form a0 + a1 + . . . + am = n X a0 a1 am gk = ca x 0 x 1 · · · x m where a1 + 2a2 + . . . + mam = k So in the ring S, we have gk =
X
ca (xq−1 q h0 )a0 (xq−1 q−1 h1 )a1 · · · (xq−1 2 hq−2 )aq−2 xq−1 aq−1 xaq q · · · xamm
=
X
q−2 aq ca xq−1 M ha00 · · · hq−2 xq · · · xamm
a
43
where M = qa0 + (q − 1)a1 + . . . + aq−1 , for k ranging from 0 to (n − 1)q − 1. The conditions on ai imply that M = qn − k + (aq+1 + 2aq+2 + . . . + (m − q)am ) ≥ q + 1 for 0 ≤ k ≤ (n − 1)q − 1. This means we can express gk = xq+1 q−1 gek for all k = 0, . . . , (n − 1)q − 1. Thus, `(Rp /Jm (X)) = `(S/(g0 , . . . , g(n−1)q−1 )) ≥ q + 1.
CHAPTER IV
Jet Schemes of Determinantal Varieties
In this chapter, we study the scheme structure of the jet schemes of determinantal varieties. We show that in general, these jet schemes are not irreducible (see Theorems IV.8 and IV.18). In the case of the determinantal variety X of matrices of rank at most one, we give a formula for the dimension of each of the components of its jet schemes (see Theorem IV.19). As an application, we compute the log canonical threshold of the pair (Ars , X) (see Corollary IV.21).
4.1
Basics of determinantal varieties
Let Xc be the set of all r × s matrices of rank at most c over a field k. Since an r × s matrix has rank at most c if and only if all of its (c + 1)-minors1 vanish, and since these minors are polynomials in the entries of the matrix, we see that Xc is a closed algebraic set in the affine space of all r × s matrices over k. Let (xij ) be an r × s matrix with distinct indeterminates as its entries, and let Ic+1 denote the ideal generated by the (c + 1)-minors of (xij ). A theorem of Hochster and Eagon [5, Theorem 7.3.1(c)] asserts that Ic+1 is a prime ideal, and therefore the set Xc can be naturally identified with the subvariety Spec k[xij ]/Ic+1 of Ars . We call Xc a determinantal variety. 1 We
define a d-minor of a matrix to be a d × d subdeterminant of that matrix.
44
45
Determinantal varieties arise naturally in algebraic geometry, for example, they are used to describe degeneracy loci of vector bundles. It is straightforward to check that the dimension of Xc is c(r + s − c). Hochster and Eagon showed that a determinantal variety Xc is Cohen-Macaulay and normal [5, Theorem 7.3.1(c)], and Svanes showed that Xc is Gorenstein if r = s [5, Theorem 7.3.6(b)]. 4.1.1
Group action
Determinantal varieties have a natural action by the group GL(r) × GL(s), where GL(r) acts on an r × s matrix (a point in Ars ) by left multiplication and GL(s) acts by right multiplication. Because the rank of a matrix is preserved by this action on Ars , the action obviously restricts to an action on the determinantal variety Xc . Moreover, the group GL(r) × GL(s) acts transitively on matrices of a fixed rank. That is, the orbit of a rank k matrix is the set of all rank k matrices. This is Ik 0 obvious since any r × s matrix of rank k can be brought into the form by 0 0 elementary row and column operations (which can be interrepted as multiplication by elementary matrices). 4.1.2
Singularities
In this section, we review some basic facts about the singularities of a determinantal variety. Proposition IV.1 shows that the singular locus of a determinantal variety is itself also a determinantal variety. Proposition IV.5 records the details of the folklore fact that determinantal varieties have rational singularities. Proposition IV.1. The singular locus of the variety Xc is precisely Xc−1 . Proof. We will first show that the r × s matrices of rank at most c − 1 are singular points of Xc , and then that those of rank exactly c are not.
46
We will use the Jacobian criterion to show the singularity of the smaller rank matrices. First, we need to understand how the Jacobian matrix looks. The variety Xc is defined by the ideal Ic+1 of (c + 1)-minors of a generic r × s matrix (xij ). Let 4 be a generator of Ic+1 . If the indeterminate xij does not appear in 4, then obviously ∂4 ∂xij
= 0. If xij appears in 4, then 4 can be obtained by expanding along the
row containing xij . Therefore,
∂4 ∂xij
is just a c-minor. This means that the Jacobian
matrix has entries that are either zero or c-minors. So if we evaluate it at any r × s matrix of rank at most c − 1, we get zero. Thus, the set Xc−1 is contained in the singular locus of Xc by the Jacobian criterion. Conversely, suppose one r × s matrix of rank exactly c is singular. Then by the action of GL(r) × GL(s) described in the last section, all r × s matrices of rank c are singular. But this would mean the whole variety Xc is singular, which is impossible since Xc is reduced (because the ideal Ic+1 is prime). Hence, the determinantal variety Xc is singular precisely along Xc−1 . Theorem IV.2. Determinantal varieties have rational singularities. By definition, a normal variety X has rational singularities if and only if it admits e → X such that Rp f∗ O e = 0 for all p > 0. Since a resolution of singularities f : X X this is a local condition, we may assume X is affine, in which case it has rational e O e ) = 0 for p > 0. singularities if and only if H p (X, X There is an alternate characterization of rational singularities due to Kempf [22, p. 50]: Proposition IV.3. Let X be a normal Cohen-Macaulay scheme over a field of chare → X be a resolution of singularities for X. Then X acteristic zero, and let π : X has rational singularities if and only if the natural inclusion π∗ ωXe ⊂ ωX is an iso-
47
morphism. The following result gives a sufficient condition for a normal Cohen-Macaulay variety to have rational singularities. Proposition IV.4. If π : Ye → Y is a small resolution (that is, the exceptional set2 of π has codimension at least two) of a normal Cohen-Macaulay variety Y , then Y has rational singularities. Proof. By Proposition IV.3, we need to show π∗ ωYe = ωY . Without loss of generality, we may assume that Y is affine. So it suffices to show π∗ ωYe (Y ) = ωY (Y ). By definition of a small resolution, we have π −1 (U ) ⊆ Ye ∼ =
π
U
⊆
Y
where Ye \ π −1 (U ) and Y \ U have codimension at least two. Since the canonical module on a normal variety is determined by its restriction to any open subset whose complement has codimension at least two, we have ωY (Y ) = ωY (U ) and ωYe (Ye ) = ωYe (π −1 (U )). Moreover, π −1 (U ) ∼ = U means that ωYe (π −1 (U )) = ωY (U ). Thus, π∗ ωYe (Y ) = ωYe (π −1 (Y )) = ωYe (Ye ) = ωY (Y ). Thus to show that determinantal varieties have rational singularities, it suffices to construct a small resolution. To this end, let Xc be the determinantal variety of r × s matrices of rank at most c, and Gr(c, s) the Grassmannian of c-dimensional subspaces in an s-dimensional vector space. Consider the incidence correspondance Xc × Gr(c, s) ⊇ W 8 8 p1
Xc
=
88 p2 88 8
{(P, Λ) | Λ contains the row space of P}
Gr(c, s)
2 The exceptional set of a birational map π : Y e → Y is the smallest closed set Z ⊆ Ye such that π| e is an Y −Z isomorphism.
48
By Proposition IV.4, we see then Theorem IV.2 follows from the following folklore proposition: Proposition IV.5. With notation as above, p1 is a small resolution of singularities. Proof. By construction, the map p1 is projective and therefore proper. Note also that p1 is bijective over the smooth locus of Xc , since a smooth point of Xc is a rank c matrix and thus determines a unique point in the Grassmannian Gr(c, s). Therefore, it remains to show that W is smooth. For this, we use the other projection p2 . We will first show that W is irreducible. Since the Grassmannian Gr(c, s) is irreducible, it suffices to show that all fibers of p2 are irreducible and have the same dimension [35, Chapter 1, §6, Theorem 8]. Let Λ be an arbitrary c-dimensional subspace of As . After a change of coordinates, we may assume that Λ is the row space of the matrix Ac , where Ac is an r × s matrix with a c × c submatrix in the upper left corner and zero entries everywhere else. Then p−1 2 (Λ) consists of all the r × s matrices whose entries in the last s − c columns are zero. In other words, ∼ rc p−1 2 (Λ) = A . Therefore, the fibers of p2 are irreducible and of the same dimension. Thus, W is irreducible. Now to show that W is smooth, we need to show that the Grassmannian Gr(c, s) has an open covering {Ui }i with p−1 2 (Ui ) smooth for all i. ucker coorLet U = U[i1 ,...,ic ] be the standard open patch of Gr(c, s) where the Pl¨ dinates indexed by i1 < · · · < ic are nonzero. Then W is irreducible implies that the −1 open subset p−1 2 (U ) is irreducible, and hence has a smooth point. Since p2 (U ) is a
GL(r)-homogeneous space, meaning that the group GL(r) acts transitively on it (by row operations), we have p−1 2 (U ) smooth. Thus, W is smooth. Finally, we check that p1 is a small resolution. For this we need to compute its exceptional set. Let Q be a singular point of Xc , say Q has rank k < c. Then p−1 1 (Q) is the set of all c-dimensional subspaces containing a given k-dimensional subspace of
49
∼ As . So p−1 1 (Q) = Gr(c − k, s − k). Since the dimension of the set of rank k matrices is k(r + s − k), the dimension of the exceptional fiber is max {k(r + s − k) + (c − k)(s − c)} = (c − 1)(r + s − c + 1) + (s − c).
0≤k k or m = 6 n , pp mn we have (0) (1) (2) (0) k[xij , xij , xij ]/ J2 (Xc ) + x(0) pp − 1 : 1 ≤ p ≤ k, xmn : m = n > k or m 6= n (0) (1) (2) (0) ∼ = k[xij , xij , xij ]/ x(0) pp − 1 : 1 ≤ p ≤ k, xmn : m = n > k or m 6= n . This means that the fiber over any singular point of rank at most c−2 is isomorphic to A2rs , and therefore dim π2−1 (Xc−2 ) = 2rs + (c − 2)(r + s − c + 2). Over the singular point Ac−1 , a surviving term of a generator of J2 (Xc ) has the form (0) (0)
(0)
(1) (1)
x11 x22 · · · xc−1,c−1 xij xkl (1)
(1)
where xij and xkl are two distinct entries in the lower right (r − c + 1) × (s − c + 1) (1)
submatrix of the r × s matrix (xpq ). So over Ac−1 , we obtain (0) (1) (2) (0) k[xij , xij , xij ]/ J2 (Xc ) + x(0) pp − 1 : 1 ≤ p ≤ c − 1, xmn : m = n ≥ c or m 6= n (1) (2) (1) ∼ = k[xij , xij ]/(2-minors of the matrix (xkl )1≤k≤r−c+1 ). 1≤l≤s−c+1
58
This implies that the fiber over Ac−1 has dimension (r + s − 2c + 1) + (rs − (r − c + 1)(s − c + 1)) + rs. As a result, the preimage of the set of rank c − 1 matrices under the map π2 has dimension r + s − 2c + 1 + 2rs − (r − c + 1)(s − c + 1) + (c − 1)(r + s − c + 1). Now, to compare dim π2−1 (Xcsing ) = max{dim π2−1 (Xc−2 ), dim π2−1 (Xc−1 \ Xc−2 )} and dim π2−1 (Xcreg ) = 3c(r + s − c), we observe that dim π2−1 (Xc−2 ) ≥ dim π2−1 (Xcreg ) if and only if (r − c − 1)(s − c − 1) ≥ 3 and dim π2−1 (Xc−1 \ Xc−2 ) ≥ dim π2−1 (Xcreg ) if and only if (r − c − 1)(s − c − 1) ≥ 2. But our hypotheses r, s ≥ c + 2 and r + s ≥ 2c + 6 are equivalent to the condition (r − c − 1)(s − c − 1) ≥ 3. This completes the proof that J2 (Xc ) is reducible with our assumptions on r, s and c. 4.2.3
Varieties of matrices of rank at most one
While the analysis on the scheme structure of the jet schemes of a general determinantal variety remains incomplete, the case when t = 1 is much better understood. This is largely due to the fact that the singular locus of this type of determinantal
59
varieties is an isolated origin and that we have a very nice description of the preimage of this singular set under the map πm . Mustat¸aˇ showed that the higher jet schemes of the determinantal variety of 2 × n matrices of rank at most one are all irreducible [29, Example 4.7]. However, this result does not hold for larger matrices. In fact, we have a complete understanding of the number of components of the jet schemes in this case, and a formula for the dimension of each of the components. Theorem IV.19. Let X be the variety of r×s matrices of rank at most one. Assume r > s ≥ 3. Then Jm (X) has precisely b m+1 c + 1 irreducible components and these 2 components have dimensions qrs + (m + 1 − 2q) dim X where q = 0, . . . , b m+1 c. In 2 crs + (m mod 2) dim X. particular, the dimension of Jm (X) is = b m+1 2 As preparation for the proof of this theorem, let us first examine the preimage of the origin under the natural projection πm : Jm (X) → X in a slightly more general context. Proposition IV.20. Let X ⊆ An be a closed subscheme defined over k by a set of homogeneous polynomials, all of the same degree d. Let πm : Jm (X) → X ⊆ An be the natural surjection. Then for m ≥ d ≥ 2, −1 πm (0) ∼ = Jm−d (X) × An(d−1) .
Proof. Let I be the defining ideal of X. Then an m-jet of X corresponds to a ring homomorphism k[x1 , . . . , xn ]/I → k[t]/(tm+1 ) (0)
(1)
(m)
xi 7→ xi + xi t + . . . + xi tm
60 (j)
where xi
(0)
−1 ∈ k are arbitrary. This m-jet lies in πm (0) if and only if x1 = · · · =
(0)
xn = 0. Thus such a map of rings gives a well-defined m-jet centered at the origin of X if and only if we have (1)
(m)
(m) m−1 f (t(x1 + . . . + x1 tm−1 ), . . . , t(x(1) )) ∈ (tm+1 ) n + . . . + xn t
for each generator f of I. Since I is generated by homogeneous degree d elements, this is equivalent to (1)
(m)
(m) m−1 f (x1 + . . . + x1 tm−1 , . . . , x(1) ) ∈ (tm+1−d ). n + . . . + xn t
But this is the same as saying that the ring map k[x1 , . . . , xn ]/I → k[t]/(tm−d+1 ) (1)
(2)
(m−d+1) m−d
xi 7→ xi + xi t + . . . + xi
t
(m−d+2)
is an (m−d)-jet of X. Since there are no constraints on the variables xi
(m)
, . . . , xi
−1 (0) ∼ for all i = 1, . . . , n, we see that πm = Jm−d (X) × An(d−1) .
Proof of Theorem IV.19. We will proceed by induction on m. Because we will use Proposition IV.20 to relate Jm to Jm−2 , we will need base cases for m = 0 and m = 1. If m = 0, then b m+1 c = 0 and J0 (X) ∼ = X. The theorem predicts one 2 component of dimension same as that of X, which is obvious. If m = 1, the closed subset π1−1 (X reg ) is an irreducible component of J1 (X) and has dimension twice that of X. Note that the singular locus of X is simply the origin by Proposition IV.1, and k[xij ]/(J1 (X) + (xij )) ∼ = k[xij ], (k)
(0)
(1)
(0) as is easy to see that J1 ⊆ (xij : 1 ≤ i ≤ r, 1 ≤ j ≤ s). So π1−1 (0) ∼ = Spec Ars . The
condition r > s ≥ 3 is equivalent to (4.2)
rs ≥ 2 dim X.
61
Thus, J1 (X) has two components: π1−1 (X reg ) of dimension 2 dim X and π1−1 (X sing ) of dimension rs, as predicted by the theorem. This completes the m = 1 base case. −1 (X reg ) is irreducible and has dimension (m + For general m, the closed subset πm −1 1) dim X. On the other hand, Proposition IV.20 tells us πm (0) ∼ = Jm−2 (X) × Ars . −1 So by induction, πm (0) has b m−1 c + 1 components, where the q th component has 2
dimension qrs + (m − 1 − 2q) dim X + rs
for q = 0, . . . , b
m−1 c. 2
−1 (0) has b m+1 Because dim X = r + s − 1, it follows that πm c components, where the 2
q th component has dimension qrs + (m + 1 − 2q) dim X
where q = 1, . . . , b
m+1 c. 2
Notice that by our assumptions on r and s, the minimum value is rs+(m−1) dim X, which is greater than or equal to the dimension of the component of Jm (X) over −1 (0) is the smooth part by (4.2). Therefore, each of these b m+1 c components of πm 2
a component of Jm (X). Thus, Jm (X) has b m+1 c + 1 components of dimensions 2 qrs + (m + 1 − 2q) dim X where q ranges from 0 to b m+1 c. 2 Corollary IV.21. With the same assumptions as in Theorem IV.19, the log canonical threshold of the pair (Ars , X) is exactly 12 rs. The log canonical threshold of a pair (X, Y ) is an invariant of birational geometry reflecting the singularities of Y in X. The idea is that the smaller the threshold, the “nastier” the singularities. The definition of the log canonical threshold requires some more notion in birational geometry which we now introduce (see also [23, §3] and [25, §9.1]).
62
Definition IV.22. Let X, Y and the datum of a log resolution be as in Definition IV.12. Let c be a non-negative rational number. We say that the pair (X, cY ) is Kawamata log terminal (klt) if and only if ki − cai > −1 for all i = 1, . . . , r. Definition IV.23. The log canonical threshold of the pair (X, Y ) is lct(X, Y ) = sup{c|(X, cY ) is klt} = sup{c|ki − cai > −1 for 1 ≤ i ≤ r} ki + 1 = min . i ai Now we proceed to proving our corollary: Proof of Corollary IV.21. By applying a result of Mustat¸ˇa [30, Corollary 0.2], we obtain dim Jm (X) m+1 m≥0 b m+1 crs + (m mod 2) dim X = rs − sup 2 m+1 m≥0 m+1 crs b = rs − sup 2 m≥0 m + 1
lct(Ars , X) = dim Ars − sup
1 = rs − rs 2 1 = rs. 2
Remark IV.24. The previous three theorems give a host of examples of Gorenstein varieties with rational singularities whose jet schemes are not irreducible. In particular, they illustrate that Mustat¸ˇa’s result that locally complete intersection varieties have rational singularities if and only if their jet schemes are irreducible cannot be weakened: we may not replace the local complete intersection hypothesis with a
63
Gorenstein hypothesis. For example, taking r = s = c + 3 ≥ 4 in Theorem IV.8 gives a rationally singular Gorenstein variety whose odd jet schemes are not irreducible. Mustat¸aˇ himself also gave an example of a toric variety to illustrate this fact [29, Example 4.6].
CHAPTER V
Truncated Wedge Schemes
In this chapter, we first show a few nice properties of truncated wedge schemes, including a functorial representation (see Proposition V.2), base change under ´etale morphisms (see Proposition V.3), smoothness of truncated wedge schemes of a smooth scheme (see Corollary V.5), and a description of the first truncated wedge scheme in terms of the first jet scheme (see Proposition V.6). We also give an irreducibility criterion for truncated wedge schemes of a locally complete intersection variety (see Theorem V.8) analogous to Mustat¸aˇ’s for jet schemes. Next we present two approaches to the reduced scheme structure of the truncated wedge schemes of monomial schemes. We explicitly calculate the minimal primes of the truncated wedge schemes of monomial hypersurfaces (see Theorem V.11) analogous to that of Goward and Smith for jet schemes. Lastly, we give evidence that the irreducible components of the truncated wedge schemes of a reduced monomial hypersurface all have multiplicity one (see Section 5.4.3).
5.1
Definition and Example
Given a scheme X of finite type over a field k, an m-wedge of X is a k-morphism Spec k[s, t]/(s, t)m+1 → X.
64
65
The collection of all m-wedges of X has a natural scheme structure, as we explain below. We call it the mth wedge scheme of X, and denote it by Wm (X). If X is the affine space Ar , then an m-wedge of X corresponds to a k-algebra homomorphism k[x1 , . . . , xr ] → k[s, t]/(s, t)m+1 (0,0)
xi 7→ xi
(m,0) m
+ . . . + xi
(m−1,1) m−1
s + xi
s
(0,m) m
t + . . . + xi
t .
(i ,j ) So the truncated wedge scheme Wm (Ar ) ∼ = Spec k[xk k k ] where 1 ≤ k ≤ r and 1
0 ≤ ik + jk ≤ m. In other words, the scheme Wm (Ar ) = A 2 r(m+1)(m+2) . Now if X ⊆ Ar is a closed subscheme, then its truncated wedge scheme Wm (X) is a closed subscheme of Wm (Ar ). Say X = Spec k[x1 , . . . , xr ]/(f1 , . . . , fd ). Then an m-wedge of X corresponds to a k-algebra homomorphism φ : k[x1 , . . . , xr ]/(f1 , . . . , fd ) → k[s, t]/(s, t)m+1 (0,0)
xi 7→ xi
(m,0) m
+ . . . + xi
(0,m) m
s + . . . + xi
t ,
subject to the relations φ(fk ) = 0. Therefore, Wm (X) is defined by the ideal Wm (X) whose generators gij are the coefficients of si tj in φ(fk ) for 1 ≤ k ≤ d, 0 ≤ i + j ≤ m. Note that this computation commutes with localization. So this local construction of truncated wedge schemes can be patched together to give a scheme structure on the set of m-wedges of X for any scheme X of finite type over k.
66
Example V.1. If X = Spec k[x, y]/(xy), then the generators of W2 (X) are g00 = x00 y00 g01 = x00 y01 + x01 y00 g10 = x00 y10 + x10 y00 g02 = x00 y02 + x01 y01 + x02 y00 g11 = x00 y11 + x01 y10 + x10 y01 + x11 y00 g20 = x00 y20 + x10 y10 + x20 y00 . m : Note that W0 (X) ∼ = X. Also, we have the natural projection maps πm−1
Wm (X) → Wm−1 (X) and πm : Wm (X) → X induced by pulling back an mwedge via the natural maps Spec k[s, t]/(s, t)m+1 ,→ Spec k[s, t]/(s, t)m and Spec k ,→ Spec k[s, t]/(s, t)m+1 .
5.2
Properties
In this section, we show that truncated wedge schemes have nice properties similar to those of jet schemes. The first example is truncated wedge schemes as representing schemes of a functor. Proposition V.2. Given a scheme X of finite type over k, the mth wedge scheme Wm (X) represents the functor F : k-Schemes → Sets Z 7→ Homk (Z ×k Spec k[s, t]/(s, t)m+1 , X). Proof. From our general discussion on functors of points in Section 2.2, our goal is to show that the functor F described in the proposition is equivalent to FWm (X) , the functor of points of Wm (X).
67
To understand the set Homk (Z ×k Spec k[s, t]/(s, t)m+1 , X), one can check that we may assume X is affine. Moreover, we can assume Z is affine by Remark II.5. Say X = Spec k[x1 , . . . , xr ]/(f1 , . . . , fd ) and Z = Spec R for some k-algebra R. An element of Homk (Z ×k Spec k[s, t]/(s, t)m+1 , X) corresponds to a k-algebra map φ : k[x1 , . . . , xr ]/(f1 , . . . , fd ) → R[s, t]/(s, t)m+1 (0,0)
xk 7→ xk (i ,jk )
where xk k
(m,0) m
+ . . . + xk
(m−1,1) m−1
s + xk
s
(0,m) m
t + . . . + xk
t
∈ R arbitrary, subject to the conditions φ(fl ) = 0 for l = 1, . . . , d.
So an element of Homk (Z ×k Spec k[s, t]/(s, t)m+1 , X) corresponds to an N -tuple (0,0)
α = (α1
(1,m−1)
, . . . , α1
(0,m)
, α1
(0,0)
, . . . , αr
(0,m)
, . . . , αr
) ∈ RN satisfying the equations
φ(fl ) = 0 for all l. But from our discussion on the scheme structure of truncated wedge schemes in Section 5.1, we see that α is an R-valued point of Wm (X). In other words, we have shown that Homk (Z ×k Spec k[s, t]/(s, t)m+1 , X) = Homk (Z, Wm (X)) for all k-schemes Z, and hence the scheme Wm (X) represents the functor F . Now using this functorial point of view, we will show that truncated wedge schemes behave well under ´etale morphisms. Proposition V.3. If f : X → Y is an ´etale morphism of finite type k-schemes, then Wm (X) ∼ = Wm (Y ) ×Y X for all m. Recall the definition of formal ´etaleness [18, Definition (17.1.1)]: Definition V.4. Let f : X → Y be a morphism of schemes. We say f is formally ´etale if for any affine scheme V , any closed subscheme V0 of W defined by a nilpotent ideal of OV , and any morphism V → Y , the map of sets HomY (V, X) → HomY (V0 , X)
68
sending a homomorphism g : V → X over Y to its restriction g|V0 : V0 → X, is bijective. We say f is ´etale if it is formally ´etale and X, Y are schemes of finite type. Proof of Proposition V.3. We show the equality on the level of the corresponding functor of points. That is, we show the two sets of k-morphisms Homk (Z, Wm (X)) = Homk (Z ×k Spec k[s, t]/(s, t)m+1 , X) and Homk (Z, Wm (Y ) ×Y X) = Homk (Z, Wm (Y )) ×Homk (Z,Y ) Homk (Z, X) = Homk (Z ×k Spec k[s, t]/(s, t)m+1 , Y ) ×Homk (Z,Y ) Homk (Z, X) are the same for all k-schemes Z. By Remark II.5, it suffices to check the equality for all affine schemes Z. Fix a k-scheme Z = Spec R and consider the commutative diagram Z α
/
Z ×k Spec k[s, t]/(s, t)m+1
l γ l l l l l l f u l
β
/Y X where the top horizontal map is the closed embedding Spec R ,→ Spec R[s, t]/(s, t)m+1 .
Obviously, a Z-valued m-wedge γ ∈ Homk (Spec R[s, t]/(s, t)m+1 , X) of X induces a Z-valued m-wedge β ∈ Homk (Spec R[s, t]/(s, t)m+1 , Y ) and a map α ∈ Homk (Spec R, X) by composition. Conversely, the maps α and β together induce a unique map γ by the definition of formal ´etaleness. So we have shown Homk (Z ×k Spec k[s, t]/(s, t)m+1 , X) = Homk (Z ×k Spec k[s, t]/(s, t)m+1 , Y ) ×Homk (Z,Y ) Homk (Z, X), or equivalently, Homk (Z, Wm (X)) = Homk (Z, Wm (Y ) ×Y X)
69
for all k-schemes Z. That is, the schemes Wm (X) and Wm (Y ×Y X) are isomorphic.
When X is a smooth scheme over k, we also have the result that Wm (X) is an affine bundle over X: Corollary V.5. Let X be a smooth scheme over k of dimension n. Then Wm (X) is 1
locally an A 2 nm(m+3) -bundle over X. In particular, Wm (X) is smooth of dimension 1 n(m 2
+ 1)(m + 2).
Proof. Since X is smooth over k, X is covered by an open affine cover {Ui }i with Ui → Vi ´etale, for some Vi open subset of An . Then Wm (Ui ) ∼ = Wm (Vi ) ×Vi Ui
(by Proposition V.3)
∼ = (Wm (An ) ×An Vi ) ×Vi Ui
(since an open immersion is ´etale)
∼ = Wm (An ) ×An Ui . 1
Now Wm (An ) is an A 2 nm(m+3) -bundle over An (see Section 5.1) implies that Wm (X) 1
is an A 2 nm(m+3) -bundle over X. We saw in Chapter I that the first jet scheme of any scheme X is the total tangent space of X. Interestingly, the first truncated wedge scheme is also related to it. Proposition V.6. Let X be any scheme of finite type over k. Then W1 (X) ∼ = J1 (X) ×X J1 (X) in the category of k-schemes. Proof. We show equality on the level of functors of points; that is, we show the two sets of k-morphisms Homk (Z, W1 (X)) = Homk (Z ×k Spec k[s, t]/(s, t)2 , X)
70
and Homk (Z, J1 (X) ×X J1 (X)) are the same for all k-schemes Z. Again, it is sufficient to check the equality for all affine schemes Z. In the category of k-Algebras, we always have coproducts and pushout squares while products and pullback squares are much less common (for definition of pushout squares and pullback squares, see [28, p. 65 and p. 71 respectively]). However, we do have the following pullback square: k[s, t]/(s, t)2
k[s, t]/(s, t2 )
/
k[s, t]/(s2 , t) /
k
where the horizontal maps are natural surjections sending t to zero, and the vertical maps are natural surjections sending s to zero. That is, the ring k[s, t]/(s, t)2 is a product in the category of k-algebras. We leave the checking to the reader, but caution that analogous diagrams for higher powers are not also pullback squares. Now we apply the contravariant functor Spec(−) to our pullback square of k-algebras, and get the commutative diagram: Spec k[s,O t]/(s, t)2 o
Spec k[s,O t]/(s2 , t)
Spec k[s, t]/(s, t2 ) o
Spec k
where both the top horizontal and the left vertical maps are closed embeddings. We will show that this diagram is a pushout square in the cateogry of k-schemes. In
71
other words, for any k-scheme Y forming a commutative diagram YQ ngO O O O O O Spec k[s, t]/(s, t)2 o
Spec k[s, t]/(s2 , t)
O
O
Spec k[s, t]/(s, t2 ) o
Spec k,
there exists a unique k-morphism Spec k[s, t]/(s, t)2 → Y making the whole diagram commutative. Since both the images of Spec k[s, t]/(s2 , t) and Spec k[s, t]/(s, t2 ) in Y are contained in some affine subscheme Y0 ⊆ Y , we can replace Y with Y0 and therefore assume Y is affine. Now the existence of the unique k-morphism Spec k[s, t]/(s, t)2 → Y is obvious because of the pullback square of k-algebras mentioned earlier. So the scheme Spec k[s, t]/(s, t)2 satisfies the universal property of a coproduct of k-schemes. Next, we apply the covariant functor Z ×k − to this pushout square of k-schemes, and one can check that we have a pushout square: Z ×k Spec k[s, t]/(s, t)2 o
Z ×k Spec k[s]/(s2 )
Z ×k Spec k[t]/(t2 ) o
Z ×k Spec k
O
O
Finally, applying the contravariant and left exact functor Homk (−, X), we have a pullback square in the category of sets: Homk (Z ×k Spec k[s, t]/(s, t)2 , X)
Homk (Z ×k Spec k[t]/(t2 ), X)
/
Homk (Z ×k Spec k[s]/(s2 ), X) /
Homk (Z, X)
On the other hand, we also have an obvious pullback square in the category of
72
k-Schemes: /
J1 (X) ×X J1 (X)
J1 (X)
/
J1 (X)
X
We apply the covariant and left exact functor Homk (Z, −) and obtain again a pullback square in the category of sets: Homk (Z, J1 (X) ×X J1 (X))
Homk (Z, J1 (X))
/
Homk (Z, J1 (X)) /
Homk (Z, X)
By the uniqueness of fiber products, we have Homk (Z × Spec k[s, t]/(s, t)2 , X) = Homk (Z, J1 (X) ×X J1 (X)) for all (affine) k-schemes Z. Therefore, W1 (X) ∼ = J1 (X) ×X J1 (X). Remark V.7. From our discussion of the scheme structure of a truncated wedge scheme in Section 5.1, it suffices to prove Proposition V.6 for X affine. In this case, we can directly check that the two coordinate rings are the same by explicitly writing down generators for the defining ideals of W1 (X) and J1 (X) ×X J1 (X).
5.3
Truncated wedge schemes of local complete intersections
In the case of a locally complete intersection variety, we give an irreducibility criterion for its truncated wedge schemes similar to that of Mustat¸aˇ for jet schemes [29, Proposition 1.4]. Theorem V.8. Let X be locally a complete intersection variety of dimension n. Then the scheme Wm (X) is pure dimensional if and only if 1 dim Wm (X) = n(m + 1)(m + 2), 2
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and in this case Wm (X) is locally a complete intersection. Similarly, Wm (X) is irreducible if and only if 1 −1 dim πm (X sing ) < n(m + 1)(m + 2), 2 where πm : Wm (X) → X are the natural projections. Proof. We have a decomposition (5.1)
−1 −1 (X reg ) (X sing ) ∪ πm Wm (X) = πm
−1 (X reg ) is an irreducible component of W (X) of dimension and in general πm m
1 n(m + 1)(m + 2) 2 by Proposition V.5. So the “only if” part of both assertions is obvious and holds without the local complete intersection hypothesis. Suppose that dim Wm (X) = 21 n(m + 1)(m + 2). Working locally, we may assume that X ⊆ AN and X is defined by N − n equations. From our discussion in Section 5.1, we see that each defining equation of X gives rise to 21 (m + 1)(m + 2) defining 1
equations of Wm (X). So Wm (X) ⊆ Wm (AN ) = A 2 N (m+1)(m+2) is defined by 1 (N − n)(m + 1)(m + 2) 2 equations. Then by Krull’s principal ideal theorem [15, Theorem 10.2], every irreducible component of Wm (X) has dimension at least 12 n(m + 1)(m + 2). Thus dim Wm (X) = 21 n(m + 1)(m + 2) implies that Wm (X) is pure dimensional and a local complete intersection. −1 Now if dim πm (X sing )
