On generic coverings of the plane
by
Vik.S. Kulikov
MIAN RAN

Every nonsingular projective surface S over C defines three underline structures

vS,         dS,         tS,

where tS is the topological type of S, dS is the underline smooth 4-manifold and vS is the deformation type of S.

The aim of the talk is to give a short survey on results being relative to a Program on investigation of smooth structures on projective surfaces and their deformation types. It consists of three parts.

The first one coincides with Chisini's Problem. Let S be a nonsingular surface in a projective space P^r of degS=N. It is well known that for almost all projections pr:P^r --> P² the restrictions f:S --> P² of these projections to S satisfy the following conditions:

1. f is a finite morphism of degf=degS;
2. f is branched along an irreducible curve B subset P² with ordinary cusps and nodes, as the only singularities;
3. f^*(B)=2R+C, where R is irreducible and non-singular, and C is reduced;
4. f _{| R}:R --> B coincides with the normalization of B.

We shall call such f a generic morphism and its branch curve will be called the discriminant curve.

Two generic morphisms (S₁, f₁), (S₂, f₂) with the same discriminant curve B are said to be equivalent if there exists an isomorphism j: S₁ --> S₂ such that f₁=f₂ o j.

The following question is known as Chisini's Problem.

Problem Let B be the discriminant curve of a generic morphism f:S --> P² of degree degf >= 5. Is f uniquely determined by the pair (P², B)?

It is easy to see that the answer to the similar question for generic morphisms of projective curves to P¹ is negative. On the other hand one can show that Chisini's Problem holds for the discriminant curves of almost all generic morphisms of any projective surface.

The second part of this Program deals with so called braid monodromy technique. Let B be an algebraic curve in P² of degree 2d, where d in [ 1/2]N (if B is a discriminant curve, then degB is even, i.e. d in N). The topology of the embedding B subset P² is determined by the braid monodromy of B which is described by a factorization of the "full twist" \Delta_2d² in the semi-group B⁺_2d of the braid group B_2d of 2d string braids (in standard generators, \Delta_2d²=(X₁·...·X_2d-1)^2d). If B is a cuspidal curve, then this factorization can be written as follows

\Delta2d2= Õ i  Qi-1X1\rhoiQi,         \rhoi in (1, 2, 3),

(1)

where X₁ is a positive half-twist in B_2d.

Let

h=g1·...·gr

(2)

be a factorization in B⁺_2d. The transformation which changes two neighboring factors in (2) as follows

gi·gi+1 --> (gigi+1gi-1)·gi,

or

gi·gi+1 --> gi+1(gi+1-1gigi+1)

is called a Hurwitz move.

For z in B_2d, we denote by

hz=z-1g1z·z-1g2z·...·z-1grz

and say that the factorization expression h_z is obtained from (2) by simultaneous conjugation by z. Two factorizations are called Hurwitz and conjugation equivalent if one can be obtained from the other by a finite sequence of Hurwitz moves followed by a simultaneous conjugation. For any algebraic curve B subset P² any two factorizations of the form (1) are Hurwitz and conjugation equivalent. We shall say that two factorizations of the form (1) belong to the same braid factorization type if they are Hurwitz and conjugation equivalent. The main problem in this direction is the following one.

Problem Does the braid factorization type of the pair (P², B) uniquely determine the homeomorphism (resp. diffeomorphism) type of this pair (P², B), and vice versa?

Let S₁ and S₂ be two non-singular projective surfaces, and let j:S₁ --> S₂ be a homeomorphism. The homeomorphism j induces the isomorphism j^*:H²(S₂, Z) --> H²(S₁, Z). Assume that L_i, i=1, 2, is an ample line bundle on S_i such that f_i:S_i --> P² given by three-dimensional linear subsystem of |L_i| is a generic morphism, and let j^*(L₂)=L₁. The third part of the Program can be formulated as the following problem.

Problem Let f_i:S_i --> P², i=1, 2, be a generic morphism as above and such that Chisini's Conjecture holds for its discriminant curve B_i. Do the diffeomorphism (resp. deformation) types of S₁ and S₂ coincide if the diffeomorphism (resp. deformation) types of the pairs (P², B₁) and (P², B₂) coincide, and vice versa?

Partly supported by RFBR (No. 99-01-01133) and INTAS-OPEN (No. 97-2072).

Date received: June 29, 1999