Margarida Mendes Lopes

On the classification of irregular surfaces of general type with nonbirational bicanonical map

The present paper is devoted to the classification of irregular surfaces of general type with pg > 3 and nonbirational bicanonical map. Our main result is that, if S is such a surface and if S is minimal with no pencil of curves of genus 2, then S is the symmetric product of a curve of genus 3, and therefore pg = q = 3 and K2 = 6. Furthermore we obtain some results towards the classification of minimal surfaces with pg = q = 3. Such surfaces have 6 < Kz < 9, and we show that Kz = 6 if and only if S is the symmetric product of a curve of genus 3. We also classify the minimal surfaces with pg = q = 3 with a pencil of curves of genus 2, proving in particular that for those one has Kz = 8.

Remarks on the bicanonical map for surfaces of general type

To the memory of our colleague and friend Mario Raimondo

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.

Numerical Godeaux surfaces with an involution

Minimal algebraic surfaces of general type with the smallest possible invariants have geometric genus zero and K 2 = 1 and are usually called numerical Godeauz surfaces. Although they have been studied by several authors, their complete classification is not known. In this paper we classify numerical Godeaux surfaces with an involution, i.e. an automorphism of order 2. We prove that they are birationally equivalent either to double covers of Enriques surfaces or to double planes of two different types: the branch curve either has degree 10 and suitable singularities, originally suggested by Campedelli, or is the union of two lines and a curve of degree 12 with certain singularities. The latter type of double planes are degenerations of examples described by Du Val, and their existence was previously unknown; we show some examples of this new type, also computing their torsion group.

THE BICANONICAL MAP OF SURFACES WITH p g = 0 AND K 2 [gt-or-equal, slanted] 7

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.

The degree of the generators of the canonical ring of surfaces of general type with pg = 0

Abstract. Upper bounds for the degree of the generators of the canonical rings of surfaces of general type were found by Ciliberto [7]. In particular it was established that the canonical ring of a minimal surface of general type with pg = 0 is generated by its elements of degree lesser or equal to 6, ([7];, Th. (3.6)). This was the best bound possible to obtain at the time, since Reiders results, [11], were not yet available. In this note, this bound is improved in some cases (Theorems (3.1), (3.2)). ¶ In particular it is shown that if K2≥ 5, or if K2≥ 2 and |2 KS| is base point free this bound can be lowered to 4. This result is proved by showing first that, under the same hypothesis, the degree of the bicanonical map is lesser or equal to 4 if K2≥ 3, (Theorem (2.1)), implying that the hyperplane sections of the bicanonical image have not arithmetic genus 0. The result on the generation of the canonical ring then follows by the techniques utilized in [7].

TRIPLE CANONICAL SURFACES OF MINIMAL DEGREE

We classify completely the surfaces of general type whose canonical map is 3-to-1 onto a surface of minimal degree in projective space. These surfaces fall into 5 distinct classes and we give explicit examples belonging to each of these classes. As far as we know, one of the examples thus constructed was unknown and it is a surface whose canonical system has two infinitely near base points.

Brill–Noether loci for divisors on irregular varieties

For a projective variety X, a line bundle L on X and r a natural number we consider the r-th Brill-Noether locus W^r(L,X):={\eta\in Pic^0(X)|h^0(L+\eta)\geq r+1}: we describe its natural scheme structure and compute the Zariski tangent space. If X is a smooth surface of maximal Albanese dimension and C is a curve on X, we define a Brill-Noether number \rho(C, r) and we prove, under some mild additional assumptions, that if \rho(C, r) is non negative then W^r(C,X) is nonempty of dimension bigger or equal to \rho(C,r). As an application, we derive lower bounds for h^0(K_D) for a divisor D that moves linearly on a smooth projective variety X of maximal Albanese dimension and inequalities for the numerical invariants of curves that do not move linearly on a surface of maximal Albanese dimension.

Continuous families of divisors, paracanonical systems and a new inequality for varieties of maximal Albanese dimension

Given a smooth complex projective variety X, a line bundle L of X an element v of H^1(O_X) and a section s in H^0(L) that deforms to first order in the direction v, we give a sufficient condition on v in terms of Koszul cohomology for this first order deformation to extend to an analytic deformation. We apply this result to improve known results on the paracanonical system of a variety of maximal Albanese dimension, due to Beauville in the case of surfaces and to Lazarsfeld-Popa in higher dimension. In particular, we prove the inequality p_g(X)>=\chi(K_X)+q(X)-1 for a variety X of maximal Albanese dimension without irregular fibrations of Albanese general type.

THE DEGREE OF THE BICANONICAL MAP OF A SURFACE WITH pg = 0.

In this note it is shown that, given a smooth minimal complex surface of general type S with pg(S) = 0, K 2 S = 3, for which the bicanonical map φ2K is a morphism, the degree of φ2K is not 3. This completes our earlier results, showing that if S is a minimal surface of general type with pg = 0, K 2 ≥ 3 such that |2K s | is free, then the bicanonical map of S can have degree 1, 2 or 4.