Rita Pardini

A connected component of the moduli space of surfaces with pg=0

Abstract Let S be a minimal surface of general type with p _ g ( S )=0 and K _ s 2 ⩾3 for which the bicanonical map ϕ : S→ P K_S 2 is a morphism. Then deg ϕ⩽4 by Mendes Lopes (Arch. Math. 69 (1997) 435–440) and if it is equal to 4 then K _ S 2 ⩽6 by Mendes Lopes and Pardini (A note on surfaces of general type with p _ g =0 and K 2 ⩾7, Pisa preprint, December 1999 (Eprint: math AG/9910074)). We prove that if K _ S 2 =6 and deg ϕ=4 then S is a Burniat surface (see Peters (Nagoya Math. J. 166 (1977) 109–119)). We show moreover that minimal surfaces with p_g=0, K 2 =6 and bicanonical map of degree 4 form a four-dimensional irreducible connected component of the moduli space of surfaces of general type.

The Severi inequality K2≥4χ for surfaces of maximal Albanese dimension

We prove the so-called Severi inequality, stating that the invariants of a minimal smooth complex projective surface of maximal Albanese dimension satisfy:

Rational Surfaces with Many Nodes

K^2_S\ge4\chi(S).

TRIPLE CANONICAL SURFACES OF MINIMAL DEGREE

We classify minimal complex surfaces of general type with p g = q = 3. More precisely, we show that such a surface is either the symmetric product of a curve of genus 3 or a free Z 2 -quotient of the product of a curve of genus 2 and a curve of genus 3. Our main tools are the generic vanishing theorems of Green and Lazarsfeld and the characterization of theta divisors given by Hacon in Corollary 3.4 of Fourier transforms, generic vanishing theorems and polarizations of abelian varieties.

Brill–Noether loci for divisors on irregular varieties

We describe smooth rational projective algebraic surfaces over an algebraically closed field of characteristic different from 2 which contain n ≥ b2 −2 disjoint smooth rational curves with self-intersection −2, where b2 is the second Betti number. In the last section this is applied to the study of minimal complex surfaces of general type with pg = 0 and K2 = 8, 9 which admit an automorphism of order 2.

Non-normal abelian covers

By a recent result of Viehweg, projective manifolds with ample canonical class have a coarse moduli space, which is a union of quasiprojective varieties.In this paper, we prove that there are manifolds with ample canonical class that lie on arbitrarily many irreducible components of the moduli; moreover, for any finite abelian group G there exist infinitely many components M of the moduli of varieties with ample canonical class such that the generic automorphism group GMis equal to G. In order to construct the examples, we use abelian covers. Let Y be a smooth complex projective variety of dimension ⋛ 2. A Galois cover f :X ↪ Y whose Galois group is finite and abelian is called an abelian cover of Y; by [Pal], it is determined by its building data, i.e. by the branch divisors and by some line bundles on Y, satisfying appropriate compatibility conditions. Natural deformations of an abelian cover are also introduced in [Pal]. In this paper we prove two results about abelian covers:first, that if the buildin...

Log-canonical pairs and Gorenstein stable surfaces with

A minimal surface of general type with pg(S) = 0 satisfies 1 ≤ K2 ≤ 9 and it is known that the image of the bicanonical map φ is a surface for K2 S ≥ 2, whilst for K 2 S ≥ 5, the bicanonical map is always a morphism. In this paper it is shown that φ is birational if K2 S = 9 and that the degree of φ is at most 2 if K2 S = 7 or K 2 S = 8. By presenting two examples of surfaces S with K2 S = 7 and 8 and bicanonical map of degree 2, it is also shown that this result is sharp. The example with K2 S = 8 is, to our knowledge, a new example of a surface of general type with pg = 0. The degree of φ is also calculated for two other known surfaces of general type with pg = 0, K 2 S = 8. In both cases the bicanonical map turns out to be birational.A minimal surface of general type with p g ( S ) = 0 satisfies 1 [les ] K 2 [les ] 9, and it is known that the image of the bicanonical map φ is a surface for K 2 S [ges ] 2, whilst for K 2 S [ges ] 5, the bicanonical map is always a morphism. In this paper it is shown that φ is birational if K 2 S = 9, and that the degree of φ is at most 2 if K 2 S = 7 or K 2 S = 8. By presenting two examples of surfaces S with K 2 S = 7 and 8 and bicanonical map of degree 2, it is also shown that this result is sharp. The example with K 2 S = 8 is, to our knowledge, a new example of a surface of general type with p g = 0. The degree of φ is also calculated for two other known surfaces of general type with p g = 0 and K 2 S = 8. In both cases, the bicanonical map turns out to be birational.