arXiv subjects

Alessandro Verra

Publications and source records attributed to Alessandro Verra.

At least 37 records · Page 2Linked to original sources

arXiv2019

Moduli of non-standard Nikulin surfaces in low genus

Primitively polarized genus $g$ Nikulin surfaces $(S,M,H)$ are of two types, that we call standard and non-standard depending on whether the lattice embedding $\mathbb{Z}[H] \oplus_{\perp} \mathbf{N} \subset \rm{Pic}(S)$ is primitive. Here $H$ is the genus $g$ polarization and $\mathbf{N}$ is the Nikulin lattice. We concentrate on the non-standard case, which only occurs in odd genus. In particular, we study the birational geometry of the moduli space of non-standard Nikulin surfaces of genus $g$ and prove its rationality for $g=7,11$ and the existence of a rational double cover of it when $g=9$. Furthermore, if $(S,M,H)$ is general in the above moduli space and $(C,M|_C)$ is a general Prym curve in $|H|$, we determine the dimension of the family of non-standard Nikulin surfaces of genus $g$ containing $(C, M|_C)$ for $3\leq g\leq 11$; this completes the study of the Prym-Nikulin map initiated in our previous work.

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arXiv2018

Edge and Fano on nets of quadrics

In a number of papers by Edge, and in a related paper by Fano, several properties are discussed of the family of scrolls of degree 8, in the complex projective space, whose plane sections are projected bicanonical models of a genus 3 curve C. This beautiful subject is implicitely related to the moduli of semistable rank two vector bundles on C with bicanonical determinant. We revisit and reconstruct this matter in modern terms with a modular point of view. This paper is the refereed version of a contribution to a volume in honor of W.L Edge, to be published by European Journal of Mathematics.

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arXiv2017

Half Nikulin surfaces and moduli of Prym curves

Let F^N_g be the moduli space of polarized Nikulin surfaces (Y,H) of genus g and let P^N_g be the moduli of triples (Y,H,C), with C in |H| a smooth curve. We study the natural map χ_g:P^N_g -> R_g, where R_g is the moduli space of Prym curves of genus g. We prove that it is generically injective on every irreducible component, with a few exceptions in low genus. This gives a complete picture of the map χ_g and confirms some striking analogies between it and the Mukai map m_g: P_g ->M_g for moduli of triples (Y,H,C), where (Y,H) is any genus g polarized K3 surface. The proof is by degeneration to boundary points of a partial compactification of F^N_g. These represent the union of two surfaces with four even nodes and effective anticanonical class, which we call half Nikulin surfaces. The use of this degeneration is new with respect to previous techniques.

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arXiv2017

Syzygies of Prym and paracanonical curves of genus 8

By analogy with Green's Conjecture on syzygies of canonical curves, the Prym-Green conjecture predicts that the resolution of a general level p paracanonical curve of genus g is natural. The Prym-Green Conjecture is known to hold in odd genus for almost all levels. Probabilistic arguments strongly suggested that the conjecture might fail for level 2 and genus 8 or 16. In this paper, we present three geometric proofs of the surprising failure of the Prym-Green Conjecture in genus 8, hoping that the methods introduced here will shed light on all the exceptions to the Prym-Green Conjecture for genera with high divisibility by 2.

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arXiv2017

The universal K3 surface of genus 14 via cubic fourfolds

Using the isomorphism between the moduli space of polarized K3 surfaces of genus 14 and the moduli space of special cubic fourfolds of discriminant 26, we establish the rationality of the universal K3 surface of genus 14. Precisely, we show that the universal K3 surface of genus 14 is a projective bundle over a certain moduli space of nodal scrolls in P^5, whose rationality we prove using a degenerate version of Mukai's structure theorem for curves of genus 8.

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arXiv2016

The universal theta divisor over the moduli space of curves

We carry out a complete birational classification of the universal theta divisor Th_g over the moduli space of curves of genus g, and show that Th_g enjoys good rationality properties for g<12, and is a variety of general type for g\geq 12. The key ingredient is an intersection-theoretic study of the universal antiramification locus of the Gauss map. We also present a complete classification of the universal symmetric product of degree g-2 over M_g.

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arXiv2015

Prym varieties and moduli of polarized Nikulin surfaces

We present a structure theorem for the moduli space R_7 of Prym curves of genus 7 as a projective bundle over the moduli space of 7-nodal rational curves. The existence of this parametrization implies the unirationality of R_7 and that of the moduli space of Nikulin surfaces of genus 7, as well as the rationality of the moduli space of Nikulin surfaces of genus 7 with a distinguished line. Using the results in genus 7, we then establish that R_8 is uniruled.

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arXiv2015

Geometry of genus 8 Nikulin surfaces and rationality of their moduli

Let S be a general complex Nikulin surface of genus 8, a geometric construction of S is given as follows. Consider a smooth 3-fold linear section T of the Grassmannian G(1,4) and the Hilbert scheme of rational normal sextic curves of T. In it consider the special family of sextics A which are also contained in the congruence of bisecant lines to a rational normal quartic curve of P^4. We show that S is biregular to a quadratic section of T containing a sextic A. In particular A admits a 1-dimensional family of bisecant lines contained in G(1,4) and 8 of them are in S. This explicit construction is then used to prove that the moduli space of genus 8 Nikulin surfaces is rational.

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arXiv2015

On the rationality of the moduli of higher spin curves in low genus

The global geometry of the moduli spaces of higher spin curves and their birational classification is largely unknown for g >= 2 and r > 2. Using quite related geometric constructions, we almost complete the picture of the known results in genus g <= 4 showing the rationality of the moduli spaces of even and odd 4-spin curves of genus 3, of odd spin curves of genus 4 and of 3-spin curves of genus 4.

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arXiv2015

On the unirationality of higher dimensional Ueno-type manifolds

We prove the unirationality of the Ueno-type manifold $X_{4,6}$. $X_{4,6}$ is the minimal resolution of the quotient of the Cartesian product $E(6)^4$, where $E(6)$ is the equianharmonic elliptic curve, by the diagonal action of a cyclic group of order 6 (having a fixed point on each copy of $E(6)$). We collect also other results, and discuss several related open questions.

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arXiv2015

The universal abelian variety over A_5

We establish a structure result for the universal abelian variety over the moduli space A_5, in terms of discriminant curves of conic bundles over a del Pezzo surface. In particular, this gives a very simple unirational parametrization of the boundary divisor of the moduli space A_6 of principally polarized abelian varieties of dimension 6. Using this we derive a lower bound on the slope of all effective divisors on A_6.

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arXiv2013

On the life and scientific work of Gino Fano

A collection of selected papers of Gino Fano, containing most of his research papers in Italian, is now available on line at the site of Biblioteca Digitale Italiana di Matematica. The address is: http://www.bdim.eu/ The realization of this collection on line is part of a joint project of Unione Matematica Italiana (UMI) and Societa' Italiana per la Matematica Applicata e Industriale (SIMAI). The present submission to arXiv is the introduction to the collection and an announcement of it as well.

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arXiv2011

Rational parametrizations of moduli spaces of curves

The paper aims to give an account, both historical and geometric, on the diverse geography of rational parametrizations of moduli spaces related to curves. It is a contribution to the book Handbook of Moduli, editors G. Farkas and I. Morrison, to be published by International Press. Refereed version.

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arXiv2011

Moduli of theta-characteristics via Nikulin surfaces

We study moduli spaces of K3 surfaces endowed with a Nikulin involution and their image in the moduli space R_g of Prym curves of genus g. We observe a striking analogy with Mukai's well-known work on ordinary K3 surfaces. Many of Mukai's results have a very precise Prym-Nikulin analogue, for instance: (1) A general Prym curve from R_g is a section of a Nikulin surface if and only if g\leq 7, g\neq 6. (2) R_7 has the structure of a Mori fibre space over the corresponding moduli space of polarized Nikulin surfaces. (3) In the case of genus next to maximal (g=6), the Prym-Nikulin locus in R_6 is an extremal divisor; the corresponding space of Nikulin surfaces has a Grassmannian GIT model. (4) The Prym-Nikulin locus in R_g can be characterized by extra syzygies of Prym-canonical curves. We then use these results to study the geometry of the moduli space S_g of even spin curves, with special emphasis on the transition case of S_8 which is a 21-dimensional Calabi-Yau variety.

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arXiv2011

The classification of universal Jacobians over the moduli space of curves

We carry out a complete birational classification of the degree g universal Jacobian P_g over the moduli space of curves, highlighting the transition cases g=10, 11. The universal Jacobian is unirational when g<10, has Kodaira dimension zero for g=10 and Kodaira dimension 19 for g=11. For g>11, the variety P_g has Kodaira dimension 3g-3, that is, the maximum allowed by Iitaka's easy addition formula for fibre spaces. In particular, we disprove the expectation that P_g and M_g have the same Kodaira dimension for all genera.

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arXiv2011

Plucker forms and the theta map

In this paper we introduce the elementary notion of Plücker form of a pair $(E,S)$, where $E$ is a vector bundle of rank $r$ on a smooth, irreducible, complex projective variety $X$ and $S \subset H^0(E)$ is a subspace of dimension $rm$. We apply this notion to the study of theta map $θ_r$ on the moduli space $SU_X(r,0)$ of semistable vector bundles of rank $r$ and trivial determinant on a curve $X$ of genus $g$. We prove that $θ_r$ is generically injective if $X$ is general and $g >> r$.

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arXiv2009

The intermediate type of certain moduli spaces of curves

A well-known principle of Mumford asserts that all moduli spaces of curves are varieties of general type, except a finite number of cases that occur for relatively small genus, when these varieties tend to be uniruled. In all known cases, the transition from uniruledness to being of general type is quite sudden. For the first time, we show that naturally defined moduli spaces of curves can have intermediate birational type. We show that the moduli space of even spin curves of genus 8 is of Calabi-Yau type. Then we prove that the 41-dimensional moduli space M_{11, 11} of 11-pointed curves of genus 11, has Kodaira dimension 19.

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arXiv2009

The rationality of the moduli space of genus four curves endowed with an order three subgroup of their Jacobian

Refereed version to appear in Michigan Mathematical Journal. A mistake in the last section of the previous version has been corrected. The new title exactly describes the main result obtained. Building on the geometry of cubic surfaces and on a theorem of Dolgachev, the rationality of the moduli space R mentioned in the title is proved. Let M be the moduli space of 6 points in the plane, modulo the natural involution induced by double-six configurations on cubic surfaces. It is proved that R is birational to a tower of locally trivial projective bundles ending onto M. The rationality of R then follows from Dolgachev's theorem that M is rational.

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