Clifford index for reduced curves

We extend the notion of Clifford index to reduced curves with planar singularities by considering rank 1 torsion free sheaves. We investigate the behaviour of the Clifford index with respect to the combinatorial properties of the curve and we show that Green's conjecture holds for certain classes of curves given by the union of two irreducible components.Comment: Prop. 3.7, prop. 4.2, Thm. 4.4 change

Scrolls containing binary curves

We study families of scrolls containing a given rational curve and families of rational curves contained in a fixed scroll via a stratification in terms of the degree of the induced map onto P^1 and we prove that there is no rational normal scroll of minimal degree and of dimension <=n/2 containing a general binary curve inP^n.Comment: arXiv admin note: substantial text overlap with arXiv:1402.578

On Clifford's theorem for singular curves

Let C be a 2-connected Gorenstein curve either reduced or contained in a smooth algebraic surface and let S be a subcanonical cluster (i.e. a 0-dim scheme such that the space H^0(C, I_S K_C) contains a generically invertible section). Under some general assumptions on S or C we show that h^0(C, I_S K_C) <= p_a(C) - deg (S)/2 and if equality holds then either S is trivial, or C is honestly hyperelliptic or 3-disconnected. As a corollary we give a generalization of Clifford's theorem for reduced curves

On varieties whose universal cover is a product of curves

We investigate a necessary condition for a compact complex manifold X of dimension n in order that its universal cover be the Cartesian product $C^n$ of a curve C = \PP^1 or \HH: the existence of a semispecial tensor $\omega$. A semispecial tensor is a non zero section $0 \neq \omega \in H^0(X, S^n\Omega^1_X (-K_X) \otimes \eta)$), where $\eta$ is an invertible sheaf of 2-torsion (i.e., \eta^2\cong \hol_X). We show that this condition works out nicely, as a sufficient condition, when coupled with some other simple hypothesis, in the case of dimension $n= 2$ or $n= 3$; but it is not sufficient alone, even in dimension 2. In the case of K\"ahler surfaces we use the above results in order to give a characterization of the surfaces whose universal cover is a product of two curves, distinguishing the 6 possible cases.Comment: 22 pages, dedicated to Sommese's 60-th birthday. Greatly improves, expands and supersedes arXiv:0803.3008, of which also corrects a mistak

Canonical rings of Gorenstein stable Godeaux surfaces

Extending the description of canonical rings from \cite{reid78} we show that every Gorenstein stable Godeaux surface with torsion of order at least $3$ is smoothable.Comment: 17 page

Computing invariants of semi-log-canonical surfaces

We describe some methods to compute fundamental groups, (co)homology, and irregularity of semi-log-canonical surfaces. As an application, we show that there are exactly two irregular Gorenstein stable surfaces with $K^2=1$, both of which have $\chi(X) = 0$ and $\mathrm{Pic}^0(X)=\mathbb{C}^*$ but different homotopy type.Comment: 17 page

Log-canonical pairs and Gorenstein stable surfaces with KX2=1K_X^2=1

We classify log-canonical pairs $(X, \Delta)$ of dimension two with $K_X+\Delta$ an ample Cartier divisor with $(K_X+\Delta)^2=1$, giving some applications to stable surfaces with $K^2=1$. A rough classification is also given in the case $\Delta=0$