ON RAMIFICATION VALUES OF A POLYNOMIAL MAPPING. ZBIGNIEW JELONEK Abstract. Let C1 , ..., Ck be affine algebraic varieties of dimension n − 1. There is a proper polynomial mapping F : Cn → Cn , such that the set of its ramification values contains hypersurfaces C1′ , ..., Ck′ , which are birationally equivalent to C1 , ..., Ck .
1. Main Result. Let F = (F1 , ..., Fn ) : Cn → Cn be a polynomial proper mapping. We say that a point y ∈ Cn is a ramification value of F if #F −1 (y) < µ(F ), where µ(F ) = (C(x1 , ...xn ) : C(F1 , ..., Fn )) denotes the geometric degree of F. Let C be an irreducible hypersurface, which is contained in the set of ramification values of F . Professor R.V. Gurjar asked, what we can say about the geometry of this hypersurface. In this note we show that the geometry of a hypersurface C can be in fact arbitrary. Theorem 1.1. Let C1 , ..., Ck be affine algebraic varieties of dimension n − 1. There is a proper polynomial mapping F : Cn → Cn , such that the set of its ramification values contains hypersurfaces C1′ , ..., Ck′ , which are birationally equivalent to C1 , ..., Ck . Proof. Of course, we can assume that n > 1 We can also assume that varieties C1 , ..., Ck ∪ are subsets of Cn . Let C = ki=1 Ci . There are regular functions z1 , ..., zn−1 on C, such that the mapping z = (z1 , ..., zn−1 ) : C → Cn−1 is finite. This implies that there is a n−1 Zariski open smooth and dense subset U ⊂ C, such that ∧i=1 dx zi ̸= 0 for x ∈ U. Take S = C \ U and let W = z(S) ⊂ Cn−1 . The set W is a hypersurface. n−1 In particular it is described by a reduced polynomial p = 0. Moreover, ∧i=1 dx zi ̸= 0 if ′ x ∈ U := {x : p ◦ z(x) ̸= 0}. Let hi = 0, be irreducible equations for Ci , i = 1, ..., n − 1 and ∏ take H = ki=1 hi . It is an easy observation, that differential forms dz1 , ..., dzn−1 and dH span the space (Tx Cn )∗ for every x ∈ U ′ . Let us go back ∑ to the mapping z = (z1 , ..., zn−1 ) : C → Cn−1 . It is finite, so for a generic linear form l = ni=1 ai xi , we have that for every indices ”j” the mapping ϕ = (z, l) : Cj → Cn is finite and birational. Let us note that ∑n−1 ′ there are functions A′i and B ′ , which are regular on U ′ , such that dl = i=1 Ai dzi + B ′ dH ′ on U . In particular, for some integer k > 0, we have the equality
p(z)k dl =
n−1 ∑
Ai dzi + BdH,
i=1
which holds on C. Take L = p(z)k l − BH. Note that the mapping ϕ = (z, L) : C → Cn is still birational on every irreducible component Cj and finite. Moreover, the differential form dL on C can be ∑ written as n−1 i=1 Di dzi , where Di are polynomial functions. Since the mapping ϕ = (z, L) is proper (on C), we have monic (with respect to coordinate functions xi ) polynomial 1991 Mathematics Subject Classification. 14 D 06, 14 Q 20. Key words and phrases. polynomial mapping, ramification points, ramification values. The author is partially supported by the KBN grant. 1
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ZBIGNIEW JELONEK
equations Wi (xi , ϕ) = 0 (on C). In particular polynomials Wi (xi , ϕ) = 0 considered as polynomials on Cn vanish on C. Take Φ : Cn ∋ x → (ϕ(x), W1 (x1 , ϕ)2 , ..., Wn (xi , ϕ)2 ) ∈ Cn × Cn . It is a proper mapping and rank dx Φ ≤ n − 1 for x ∈ C. Now take a generic projection π : Φ(Cn ) → Cn × {0} and put F = π ◦ Φ. Then F is proper, ramifies on C and its restriction to Ci is a birational mapping (= ϕ). To finish it is enough to take Ci′ = F (Ci ). 2. Extensions. The proof of Theorem 1.1 suggests the following generalization of Proposition 21 from [1] (see also [2]): Theorem 2.1. Let X be an affine variety and Y ⊂ X be a Zariski closed subset of X. Let G : X → Cn be a polynomial mapping, such that the restriction g : Y ∋ y → G(y) ∈ Cn is a finite mapping. Assume that dim X ≤ n. Then there exists a finite polynomial mapping F : X → Cn , such that 1) F (y) = G(y) for y ∈ Y, 2) dy F = dy G for y ∈ Y. Proof. Let G = (G1 , . . . , Gn ) and let x1 , . . . , xm be polynomial functions generating C[X] over C, i.e., C[X] = C[x1 , . . . , xm ]. Since the mapping g is finite (on Y ), we have monic (with respect to coordinate functions xi ) polynomial equations Wi (xi , g) = 0 (on Y ). In particular polynomials Wi (xi , G) = 0 considered as polynomials on X vanish on Y. Take Φ : X ∋ x → (G(x), W1 (x1 , G)2 , ..., Wm (xi , G)2 ) ∈ Cn × Cm . By the construction the mapping Φ is finite, Φ(Y ) ⊂ Cn × {0} and dim Φ(X) = dim X ≤ n. Moreover, on Y we have dx (Φ) = (dx G, 0). Let π : Φ(X) → Cn × {0} be a generic linear projection and let F = π ◦ Φ. Then the mapping F is finite and F (y) = G(y), dy F = dy G, for y ∈ Y. In Theorem 2.1 we can replace the word ”finite” by the word ”quasi-finite”. Indeed, we have: Theorem 2.2. Let X be an affine variety and Y ⊂ X be a Zariski closed proper subset of X. Let G : X → Cn be a polynomial mapping. Assume that dim X ≤ n. Then there exists a generically-finite polynomial mapping F : X → Cn , such that 1) F (y) = G(y) for y ∈ Y, 2) dy F = dy G for y ∈ Y. Moreover, if the restriction g : Y ∋ y → G(y) ∈ Cn is a quasi-finite mapping, then so is F. Proof. Let I(Y ) = (h1 , ..., hr ) denote the ideal of Y in C[X]. Let G = (G1 , . . . , Gn ). Let x1 , . . . , xm be polynomial functions generating C[X] over C, i.e., C[X] = C[x1 , . . . , xm ]. Take Hij := hi · xj ; i = 1, ..., r, j = 1, ..., m; and consider the map 2 2 2 2 H = (G1 , . . . , Gn , h21 , ..., h2r , H11 , ..., H1m , ..., Hr1 , . . . , Hrm ) : X → Cn × Cr+mr .
By the construction the mapping H is quasi-finite outside Y. Moreover, we have H(Y ) ⊂ Cn × {0} and dim H(X) = dim X ≤ n and dH = (dG, 0) on Y. Take Γ := cl(H(X)) and consider a generic linear projection π : Γ → Cn × {0}. Now take F = π ◦ H. Then the mapping F is generically-finite, F (y) = G(y) and dy F = dy G, for y ∈ Y. Moreover, if the restriction g : Y ∋ y → G(y) ∈ Cn is a quasi-finite mapping, then so is F.
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Acknowledgements. The author wishes to express his thanks to Professor R. V. Gurjar for suggesting the problem. References [1] Z. Jelonek, The set of points at which the polynomial mapping is not proper, Ann. Polon. Math. 58, 259-266, (1993), [2] Z. Jelonek, Testing sets for properness of polynomial mappings, Math. Ann. 315 (1999), 1-35. ´ (Z. Jelonek) Instytut Matematyczny, Polska Akademia Nauk, Sw. ´ w, Poland Krako E-mail address:
[email protected]
Tomasza 30, 31-027
