ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

4. Algebraic Geometry

1991 MS Classification: 14, 32 Kulikov, Viktor S., Steklov Mathematical Institute, Russia On Chisini’s Conjecture Let B ⊂ ℙ^{2} be an irreducible curve over ℂ with c ordinary cusps and n nodes, as the only singularities. Put deg B = 2d, and let g be the genus of its desingularization. The curve B is the discriminant curve of a generic morphism f : S → ℙ^{2}, deg f ≥ 3, if: (i) S is a non-singular irreducible projective surface; (ii) f is unramified over ℙ^{2} \ B; (iii) f^{∗}(B) = 2R + C, where R is irreducible and non-singular, and C is reduced; (iv) f_{|R} : R → B is the normalization of B. Chisini’s Conjecture asserts that for the discriminant curve B of a morphism f of deg f ≥ 5 this generic morphism is unique up to equivalence. We prove that if B is the discriminant curve of a morphism f of deg f > 4(3d + g − 1)/(2(3d + g − 1) − c), then, for B, the generic morphism f is unique, i. e., Chisini’s Conjecture holds for B. As a consequence we prove that if the canonical bundle K_{S} of S is ample, f is a generic morphism such that f^{−1}(ℙ^{1}) ≡ mK_{S}, m ∈ ℕ, then, for B, the generic morphism f is unique except, possibly, for 3 cases. In all the exceptional cases, if for B there exist non-equivalent generic morphisms, then these morphisms have degree ≤ 6.