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Braid monodromy factorizations and diffeomorphism types. (English. Russian original) Zbl 1004.14005

Izv. Math. 64, No. 2, 311-341 (2000); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 64, No. 2, 89-120 (2000).
From the introduction: Let \(S\subset \mathbb{C}\mathbb{P}^r\) be a non-singular algebraic surface of degree \(\deg S=N\). It is well known that for almost all projections \(\text{pr}:\mathbb{C} \mathbb{P}^r\to \mathbb{C}\mathbb{P}^2\) the restrictions \(f:S \to \mathbb{C}\mathbb{P}^2\) of these projections to \(S\) satisfy the following conditions:
(i) \(f\) is a finite morphism of degree \(\deg f=\deg S\),
(ii) \(f\) is branched along an irreducible curve \(B\subset\mathbb{C} \mathbb{P}^2\) whose singularities are ordinary cusps and nodes only,
(iii) \(f^*(B)= 2R+C\), where the curve \(R\) is irreducible and non-singular and \(C\) is reduced,
(iv) \(f|_R: R\to B\) coincides with the normalization of \(B\).
We call such an \(f\) a generic morphism, and its branch curve \(B\) is called the discriminant curve of \(f\). Two generic morphisms \((S_1,f_1)\), \((S_2,f_2)\) with the same discriminant curve \(B\) are said to be equivalent if there is an isomorphism \(\varphi: S_1\to S_2\) such that \(f_1=f_2\circ \varphi\). The following assertion is known as “Chisini’s conjecture”:
Let \(B\) be the discriminant curve of a generic morphism \(f:S\to \mathbb{C}\mathbb{P}^2\) of degree \(\deg f\geq 5\). Then \(f\) is uniquely determined by the pair \((\mathbb{C}\mathbb{P}^2,B)\).
It is easy to see that the analogous conjecture for generic morphisms of projective curves to \(\mathbb{C} \mathbb{P}^1\) is not true. On the other hand, Chisini’s conjecture holds for the discriminant curves of almost all generic morphisms of any projective surface. In particular, if \(S\) is any surface of general type with ample canonical class, then Chisini’s conjecture holds for the discriminant curves of those generic morphisms \(f:S\to \mathbb{C}\mathbb{P}^2\) that are given by a three-dimensional linear subsystem of the \(m\)-canonical class of \(S\), where \(m\in\mathbb{N}\). The discriminant curves of such generic morphisms will be called \(m\)-canonical discriminant curves.
Let \(B\) be an algebraic curve of degree \(p\) in \(\mathbb{C}\mathbb{P}^2\). The topology of the embedding \(B\subset\mathbb{C} \mathbb{P}^2\) is determined by the braid monodromy of \(B\), which is described by a factorization of the “full twist” \(\Delta^2_p\) in the semi-group \(B_p^+\) of the braid group \(B_p\) on \(p\) strings. (In the standard generators, \(\Delta^2_p=(X_1\cdot\dots\cdot X_{p-1})^p.)\) If \(B\) is a cuspidal curve, then this factorization can be written as \(\Delta^2_p= \prod_iQ_i^{-1} X_1^{\rho_i}Q_i\), \(\rho_i\in(1,2,3)\), where \(X_1\) is the positive half-twist in \(B_p\).
Main problems:
Problem 1. Let \(B\subset\mathbb{C} \mathbb{P}^2\) be a cuspidal curve. Does the braid factorization type of the pair \((\mathbb{C}\mathbb{P}^2,B)\) uniquely determine the diffeomorphism type of this pair, and vice versa?
Problem 2. Let \(\Delta^2_p= {\mathcal E}_1\) and \(\Delta^2_p= {\mathcal E}_2\) be two braid monodromy factorizations. Does there exist a finite algorithm to recognize whether these two braid monodromy factorizations belong to the same braid factorization type?
One of the main results of this paper is the following theorem.
Theorem 1. Let \(B_1,B_2\subset\mathbb{C}\mathbb{P}^2\) be cuspidal algebraic curves. Suppose that the pairs \((\mathbb{C}\mathbb{P}^2,B_1)\) and \((\mathbb{C}\mathbb{P}^2, B_2)\) have the same braid factorization type. Then the pairs \((\mathbb{C} \mathbb{P}^2,B_1)\) and \((\mathbb{C} \mathbb{P}^2,B_2)\) are diffeomorphic.
It is well known that there exist four-dimensional smooth manifolds which are homeomorphic but not diffeomorphic. One of the most important problems of four-dimensional geometry is to find invariants that distinguish smooth structures on the same topological four-dimensional manifold.
Theorem 2. Let \(f_1:S_1\to\mathbb{C} \mathbb{P}^2\) and \(f_2:S_2\to \mathbb{C}\mathbb{P}^2\) be generic morphisms of non-singular projective surfaces, and let \(B_1,B_2\subset \mathbb{C}\mathbb{P}^2\) be their discriminant curves. Suppose that Chisini’s conjecture holds for \((\mathbb{C}\mathbb{P}^2,B_1)\). If the pairs \((\mathbb{C}\mathbb{P}^2,B_1)\) and \((\mathbb{C}\mathbb{P}^2,B_2)\) have the same braid factorization type, then \(S_1\) and \(S_2\) are diffeomorphic.
Corollary. Let \(S_1,S_2\) be surfaces of general type with ample canonical class, and let \(B_1, B_2\) be \(m\)-canonical discriminant curves of generic morphisms \(f_1:S_1\to \mathbb{C} \mathbb{P}^2\) and \(f_2:S_2\to \mathbb{C}\mathbb{P}^2\) (respectively) given by three-dimensional linear subsystems of the \(m\)-canonical class of \(S_i\), where \(m\in \mathbb{N}\). If the pairs \((\mathbb{C}\mathbb{P}^2, B_1)\) and \((\mathbb{C}\mathbb{P}^2, B_2)\) have the same braid factorization type, then \(S_1\) and \(S_2\) are diffeomorphic.

MSC:

14H50 Plane and space curves
57R50 Differential topological aspects of diffeomorphisms
14E20 Coverings in algebraic geometry
14F35 Homotopy theory and fundamental groups in algebraic geometry
20F36 Braid groups; Artin groups
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