Gilberto Bini

Moduli of curves and spin structures via algebraic geometry

GILBERTO BINI AND CLAUDIO FONTANARIAbstract. Here we investigate some birational properties of two collec-tions of moduli spaces, namely moduli spaces of (pointed) stable curvesand of (pointed) spin curves. In particular, we focus on vanishings ofHodge numbers of type (p,0) and on computations of the Kodaira di-mension. Our methods are purely algebro-geometric and rely on aninduction argument on the number of marked points and the genus ofthe curves (cf. [3]).

On the Birational Geometry of the Universal Picard Variety

We compute the Kodaira dimension of the univer- sal Picard variety Pd,g parameterizing line bundles of degree d on curves of genus g under the assumption that (d−g+1,2g−2) = 1. We also give partial results for arbitrary degrees d and we investi- gate for which degrees the universal Picard varieties are birational.

Euler characteristics of moduli spaces of curves

Let M n be the moduli space of n-pointed Riemann surfaces of genus g. Denote by M n g the Deligne-Mumford compactification of M n. In the present paper, we calculate the orbifold and the ordinary Euler characteristic of M n g for any g and n such that n > 2 2g.

Mirror quintics, discrete symmetries and Shioda maps

In a recent paper, Doran, Greene and Judes considered one parameter families of quintic threefolds with finite symmetry groups. A surprising result was that each of these six families has the same Picard Fuchs equation associated to the holomorphic 3-form. In this paper we give an easy argument, involving the family of Mirror Quintics, which implies this result. Using a construction due to Shioda, we also relate certain quotients of these one parameter families to the family of Mirror Quintics. Our constructions generalize to degree n Calabi Yau varieties in (n-1)-dimensional projective space.

New examples of Calabi-Yau threefolds and genus zero surfaces

We classify the subgroups of the automorphism group of the product of four projective lines admitting an invariant anticanonical smooth divisor on which the action is free. As a first application, we describe new examples of Calabi–Yau 3-folds with small Hodge numbers. In particular, the Picard number is 1 and the number of moduli is 5. Furthermore, the fundamental group is nontrivial. We also construct a new family of minimal surfaces of general type with geometric genus zero, K2 = 3 and fundamental group of order 16. We show that this family dominates an irreducible component of dimension 4 of the moduli space of the surfaces of general type.

A Combinatorial Algorithm Related to the Geometry of the Moduli Space of Pointed Curves

As pointed out in Arbarello and Cornalba (J. Alg. Geom.5 (1996), 705–749), a theorem due to Di Francesco, Itzykson, and Zuber (see Di Francesco, Itzykson, and Zuber, Commun. Math. Phys.151 (1993), 193–219) should yield new relations among cohomology classes of the moduli space of pointed curves. The coefficients appearing in these new relations can be determined by the algorithm we introduce in this paper.

Geometric invariant theory for polarized curves

Introduction.- Singular Curves.- Combinatorial Results.- Preliminaries on GIT.- Potential Pseudo-stability Theorem.- Stabilizer Subgroups.- Behavior at the Extremes of the Basic Inequality.- A Criterion of Stability for Tails.- Elliptic Tails and Tacnodes with a Line.- A Strati_cation of the Semistable Locus.- Semistable, Polystable and Stable Points (part I).- Stability of Elliptic Tails.- Semistable, Polystable and Stable Points (part II).- Geometric Properties of the GIT Quotient.- Extra Components of the GIT Quotient.- Compacti_cations of the Universal Jacobian.- Appendix: Positivity Properties of Balanced Line Bundles.

A-Codes from Rational Functions over Galois Rings

In this paper, we describe authentication codes via (generalized) Gray images of suitable codes over Galois rings. Exponential sums over these rings help determine—or bound—the parameters of such codes.

A remark on the rational cohomology of\(\bar S_{1,n} \)

We focus on the rational cohomology of Cornalba’s moduli space of spin curves of genus 1 withn marked points. In particular, we show that both its first and its third cohomology group vanish and the second cohomology group is generated by boundary classes.

ON THE COHOMOLOGY OF

Here we investigate rational cohomology of the moduli space of stable maps to the complex projective line with a purely algebro-pgeometric approach. In particular, we prove vanishing theorems for all its odd Betti numbers, and we give an explicit description by generators and relations of its second cohomology group.