Fibrations of low genus, I

In the present paper we consider fibrations f: S \ra B of an algebraic surface onto a curve $B$, with general fibre a curve of genus $g$. Our main results are: 1) A structure theorem for such fibrations in the case $g=2$ 2) A structure theorem for such fibrations in the case $g=3$ and general fibre nonhyperelliptic 3) A theorem giving a complete description of the moduli space of minimal surfaces of general type with $K^2_S = 3, p_g = q=1$, showing in particular that it has four unirational connected components 4) some other applications of the two structure theorems.Comment: 50 pages, to appear on Annales Scientifiques de l'Ecole Normale Superieur

Product-Quotient Surfaces: new invariants and algorithms

In this article we suggest a new approach to the systematic, computer-aided construction and to the classification of product-quotient surfaces, introducing a new invariant, the integer gamma, which depends only on the singularities of the quotient model X=(C_1 x C_2)/G. It turns out that gamma is related to the codimension of the subspace of H^{1,1} generated by algebraic curves coming from the construction (i.e., the classes of the two fibers and the Hirzebruch-Jung strings arising from the minimal resolution of singularities of X). Profiting from this new insight we developped and implemented an algorithm which constructs all regular product-quotient surfaces with given values of gamma and geometric genus in the computer algebra program MAGMA. Being far better than the previous algorithms, we are able to construct a substantial number of new regular product-quotient surfaces of geometric genus zero. We prove that only two of these are of general type, raising the number of known families of product-quotient surfaces of general type with genus zero to 75. This gives evidence to the conjecture that there is an effective bound of the form gamma < Gamma(p_g,q).Comment: 33 pages, 3 figure

Chisini's conjecture for curves with singularities of type xn=ymx^n=y^m

This paper is devoted to a very classical problem that can be summarized as follows: let S be a non singular compact complex surface, f:S --> P^2 a finite morphism having simple branching, B the branch curve: to what extent does B determine f? The problem was first studied by Chisini who proved that B determines S and f, assuming B to have only nodes and cusps as singularities, the degree d of f to be greater than 5, and a very strong hypothesis on the possible degenerations of B, and posed the question if the first or the third hypothesis could be weakened. Recently Kulikov and Nemirovski proved the result for d >= 12, and B having only nodes and cusps as singularities. In this paper we weaken the hypothesis about the singularities of B: we generalize the theorem of Kulikov and Nemirovski for B having only singularities of type {x^n=y^m}, in the additional hypothesis of smoothness for the ramification divisor. Moreover we exhibit a family of counterexamples showing that our additional hypothesis is necessary.Comment: 27 pages, LaTeX, 1 figure (file=polypicc.eps