Severi varieties and branch curves of abelian surfaces of type (1,3)

Let (A,L) be a principally polarized abelian surface of type (1,3). The linear system |L| defines a 6:1 covering of A onto P2, branched along a curve B of degree 18 in P2. The main result of the paper is that for general (A,L) the curve B irreducible, admits 72 cusps, 36 nodes or tacnodes, each tacnode counting as two nodes, 72 flexes and 36 bitangents. The main idea of the proof is to use the fact that for a general (A,L) of type (1,3) the closure of the Severi variety V in |L| is dual to the curve B in the sense of projective geometry. We investigate V and B via degeneration to a special abelian surface.Comment: 17 page

Excess dimension for secant loci in symmetric products of curves

We extend a result of W. Fulton, J. Harris and R. Lazarsfeld to secant loci in symmetric products of curves. We compare three secant loci and prove the the dimensions of bigger loci can not be excessively larger than the dimension of smaller loci.Comment: final version, to appear in Collectanea Mat

The locus of points of the Hilbert scheme with bounded regularity

In this paper we consider the Hilbert scheme $Hilb_{p(t)}^n$ parameterizing subschemes of $P^n$ with Hilbert polynomial $p(t)$, and we investigate its locus containing points corresponding to schemes with regularity lower than or equal to a fixed integer $r'$. This locus is an open subscheme of $Hilb_{p(t)}^n$ and, for every $s\geq r'$, we describe it as a locally closed subscheme of the Grasmannian $Gr_{p(s)}^{N(s)}$ given by a set of equations of degree $\leq \mathrm{deg}(p(t))+2$ and linear inequalities in the coordinates of the Pl\"ucker embedding.Comment: v2: new proofs relying on the functorial definition of the Hilbert scheme. v3: Sections reorganized, new self-contained proof of the representability of the Hilbert functor with bounded regularity (Section 6

On the hypersurface of Luroth quartics

The hypersurface of Luroth quartic curves inside the projective space of plane quartics has degree 54. We give a proof of this fact along the lines outlined in a paper by Morley, published in 1919. Another proof has been given by Le Potier and Tikhomirov in 2001, in the setting of moduli spaces of vector bundles on the projective plane. Morley's proof uses the description of plane quartics as branch curves of Geiser involutions and gives new geometrical interpretations of the 36 planes associated to the Cremona hexahedral representations of a nonsingular cubic surface

The curve of lines on a prime Fano threefold of genus 8

We show that a general prime Fano threefold X of genus 8 can be reconstructed from the pair $(\Gamma,L)$, where $\Gamma$ is its Fano curve of lines and $L=O_{\Gamma}(1)$ is the theta-characteristic which gives a natural embedding \Gamma \subset \matbb{P}^5.Comment: 24 pages, misprints corrected, to appear in International Journal of Mathematic

PALEORADIOLOGICAL STUDY ON TWO INFANTS DATED TO THE 17th AND 18th CENTURIES

During an excavation campaign in the Church of the Conversion of Saint Paul in Roccapelago (North Italy), a hidden crypt was discovered, which yielded the remains of more than 400 individuals. The crypt was used as a cemetery by the inhabitants of the village of Roccapelago between the 16th and 18th centuries. Along the north side of the crypt, an area apparently separated from the rest of the burials was found, bordered by stones, where several burials of newborns and infants were concentrated. From here, five fabric rolls containing bones were recovered, and it was decided not to carry out destructive analyses, allocating the two best examples to a thorough radiological investigation to try to define the type of burial and the complete biological profile of the infant. The two rolls, subjects of this study, can be dated archaeologically between the 17th and 18th centuries. CT analysis shows a varied group of bones with a fairly good state of conservation. The paleoradiological study carried out had the primary objective of avoiding the destruction of the two rolls, ensuring their conservation; but at the same time, providing essential data to understand their nature, defining the biological profile and the type of deposition

Chen-Ruan cohomology of ADE singularities

We study Ruan's \textit{cohomological crepant resolution conjecture} for orbifolds with transversal ADE singularities. In the $A_n$-case we compute both the Chen-Ruan cohomology ring $H^*_{\rm CR}([Y])$ and the quantum corrected cohomology ring $H^*(Z)(q_1,...,q_n)$. The former is achieved in general, the later up to some additional, technical assumptions. We construct an explicit isomorphism between $H^*_{\rm CR}([Y])$ and $H^*(Z)(-1)$ in the $A_1$-case, verifying Ruan's conjecture. In the $A_n$-case, the family $H^*(Z)(q_1,...,q_n)$ is not defined for $q_1=...=q_n=-1$. This implies that the conjecture should be slightly modified. We propose a new conjecture in the $A_n$-case which we prove in the $A_2$-case by constructing an explicit isomorphism.Comment: This is a short version of my Ph.D. Thesis math.AG/0510528. Version 2: chapters 2,3,4 and 5 has been rewritten using the language of groupoids; a link with the classical McKay correpondence is given. International Journal of Mathematics (to appear

Deformation of canonical morphisms and the moduli of surfaces of general type

In this article we study the deformation of finite maps and show how to use this deformation theory to construct varieties with given invariants in a projective space. Among other things, we prove a criterion that determines when a finite map can be deformed to a one--to--one map. We use this criterion to construct new simple canonical surfaces with different $c_1^2$ and $\chi$. Our general results enable us to describe some new components of the moduli of surfaces of general type. We also find infinitely many moduli spaces $\mathcal M_{(x',0,y)}$ having one component whose general point corresponds to a canonically embedded surface and another component whose general point corresponds to a surface whose canonical map is a degree 2 morphism.Comment: 32 pages. Final version with some simplifications and clarifications in the exposition. To appear in Invent. Math. (the final publication is available at springerlink.com

The Noether\u2013Lefschetz locus of surfaces in toric threefolds

The Noether-Lefschetz theorem asserts that any curve in a very general surface (Formula presented.) in (Formula presented.) of degree (Formula presented.) is a restriction of a surface in the ambient space, that is, the Picard number of (Formula presented.) is (Formula presented.). We proved previously that under some conditions, which replace the condition (Formula presented.), a very general surface in a simplicial toric threefold (Formula presented.) (with orbifold singularities) has the same Picard number as (Formula presented.). Here we define the Noether-Lefschetz loci of quasi-smooth surfaces in (Formula presented.) in a linear system of a Cartier ample divisor with respect to a (Formula presented.)-regular, respectively 0-regular, ample Cartier divisor, and give bounds on their codimensions. We also study the components of the Noether-Lefschetz loci which contain a line, defined as a rational curve which is minimal in a suitable sense