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Alessandro Verra

Publications and source records attributed to Alessandro Verra.

At least 19 recordsLinked to original sources
arXiv2026

On Fano varieties of index one that are linear sections of Grassmannians of lines

In this paper we consider the general $n-2$ dimensional linear section $X$ of the Grassmannian $\mathbb G(1,n)$ of lines in $\mathbb{P}^n$, that is a Fano variety of index 1. We first prove that for $n$ odd $X$ is birational to a hypersurface defined by a pfaffian determinant of order $n+1$ of linear forms in $\mathbb{P}^{n-1}$, and we study this birational transformation in some detail. Then for $n$ even, we prove that $X$ is unirational and birational to a hypersurface defined by a pfaffian determinant of order $n$ of linear forms in $\mathbb{P}^{n-1}$. Moreover we prove that the hypersurface of $\mathbb{P}^n$ described by the lines corresponding to the points in $X$ has degree $n-1$ and is rational, and we find a rational parametrization of it.

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arXiv2026

The geometry of the moduli space of odd spin curves

We describe the birational geometry of the moduli space S_g^{-} of odd spin curves (theta-characteristics) for all genera g. The odd spin moduli space is a uniruled variety for g<12, and of general type for g at least 12. Furthermore, for g<9 we use the existence of Mukai models of the moduli space of curves, to prove that S_g^{-} is unirational. Our results show that in genus 8, the odd spin moduli space in unirational, whereas its even counterpart is of Calabi-Yau type.

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arXiv2026

The unirationality of $S_9^-$ and moduli spaces of pointed spin curves

We show that the moduli space of odd spin curves of genus 9 is unirational. This is the highest genus for which such a result is known. This is achieved by realizing birationally the moduli space of odd spin curves of genus g<10 as a locally trivial projective bundle over a certain (finite quotient of the) moduli space of n-pointed odd stable spin curves of genus g'<g. We then present general results on the Kodaira dimension of both components of the moduli spaces of n-pointed spin curves of genus g.

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arXiv2026

Rational elliptic surfaces with six singular double fibres

A rational elliptic surface with section is a smooth, rational, complex, projective surface $\mathcal{X}$ that admits a relatively minimal fibration $f: \mathcal{X}\longrightarrow \bbP^1$ such that its general fibre is a smooth irreducible curve of genus one and $f$ has a section. In this paper, we classify rational elliptic surfaces with section that have exactly six singular fibres, each counted with multiplicity two. The fibres that appear with multiplicity exactly two are either of type $II$ or of type $I_2$ of the Kodaira classification. We interpret our classification from various viewpoints: a pencil of plane cubic curves, the Weierstrass equation, a double cover of $\bbF_2$ branched over an appropriate trisection of the ruling of $\bbF_2$ plus the negative section, a double cover of the plane branched along a quartic curve, plus the datum of a point on the plane. Moreover, either we give explicit normal forms for the plane quartic curve, or we indicate how to find it.

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arXiv2025

Varieties of Nodal surfaces, coding theory and Discriminants of cubic hypersurfaces. Part 1: Generalities and nodal K3 surfaces. Part 2: Cubic Hypersurfaces, associated discriminants. Part 3: Nodal quintics. Part 4: Nodal sextics

We attach two binary codes to a projective nodal surface (the strict code K and, for even degree d, the extended code K' ) to investigate the `Nodal Severi varieties F(d, n) of nodal surfaces in P^3 of degree d and with n nodes, and their incidence hierarchy, relating partial smoothings to code shortenings. Our first main result solves a question which dates back over 100 years: the irreducible components of F(4, n) are in bijection with the isomorphism classes of their extended codes K', and these are exactly all the 34 possible shortenings of the extended Kummer code K' , and a component is in the closure of another if and only if the code of the latter is a shortening of the code of the former. We extend this result classifying the irreducible components of all nodal K3 surfaces in the same way, and we fully classify their extended codes. In this classification there are some sporadic cases, obtain through projection from a node. For surfaces of degree d=5 in P^3 we determine (with one possible exception) all the possible codes K, and for several cases of K, we show the irreducibility of the corresponding open set of F(5, n), for instance we show the irreducibility of the family of Togliatti quintic surfaces. In the fourth part we show that a `Togliatti-like' description holds for surfaces of degree 6 with the maximum number of nodes= 65: they are discriminants of cubic hypersurfaces in P^6 with 31 (respectively 32) nodes, and we have an irreducible 18-dimensional family of them. For degree d=6, our main result is based on some novel auxiliary results: 1) the study of the half-even sets of nodes on sextic surfaces, 2) the investigation of discriminants of cubic hypersurfaces X, 3) the computer assisted proof that, for n = 65, both codes K, K' are uniquely determined, 4) the description of these codes, relating the geometry of the Barth sextic with the Doro-Hall graph.

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arXiv2024

On the variety of tritangential planes to a general K3 surface of degree 6 and genus 4 in $\mP^4$

Let $S\subset \mP^4$ be a general K3 surface of degree 6 and genus 4. In this paper we study the irreducible variety $X_S$ of \emph{tritangential planes} to $S$ whose general point is a plane that intersects $S$ in a curvilinear scheme of length six supported at three non collinear points. The variety $X_S$ can be identified as the relevant part of the fixed locus of the so called \emph{Beauville involution} defined on the Hilbert scheme $S[3]$ of 0--dimensional schemes of length three of $S$. In this paper we prove that: (a) $X_S$ has dimension 3, is irreducible and smooth, except for 210 points that are at most of multiplicity 2 for $X_S$; (b) $X_S$, in its natural embedding in the Grassmannian $\mathbb G(2,4)\subset \mP^9$ of planes in $\mP^4$, has degree $152$.

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arXiv2023

On deformations of the surfaces of bitangents to smooth quartic surfaces in $\mP^3$

We prove that the surface $S(X)$ of bitangent lines of a general smooth quartic surface $X$ in $\mP^3$ has unobstructed deformations of dimension $20=h^1(S(X), T_{S(X)})$. In addition, we show that the space of infinitesimal embedded deformations of $X$ injects into the one of $S(X)$. Finally we prove that there is a natural birational map from the 20--dimensional moduli space of (polarised) double coverings of EPW--sextics to the moduli space of regular surfaces $S$ with $p_g=45$ and $K_S^2=360$ polarised with a very ample line bundle $H$ such that $H^2=40$, $h^0(S, H)=6$: the map sends a double covering of a EPW--sextic in $\mP^5$ to the surface of double points of the EPW--sextic.

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arXiv2022

From Enriques surface to Artin-Mumford counterexample

After an Introduction to the themes of Enriques surfaces and Rationality questions, the Artin-Mumford counterexample to Lueroth problem is revisited. A construction of it is given, which is related in an explicit way to the geometry of Enriques surfaces, more precisely to the special family of Reye congruences and their classical geometry.

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arXiv2022

Singularities of theta divisors and the geometry of A_5

We study the codimension two locus H in A_g consisting of principally polarized abelian varieties whose theta divisor has a singularity that is not an ordinary double point. We compute the class of H in A_g for every g. For g=4, this turns out to be the locus of Jacobians with a vanishing theta-null. For g=5, via the Prym map we show that H in A_5 has two components, both unirational, which we completely describe. This gives a geometric classification of 5-dimensional ppav whose theta-divisor has a quadratic singularity of non-maximal rank. We then determine the slope of the effective cone of A_5 and show that the component N_0' of the Andreotti-Mayer divisor has minimal slope 54/7. Furthermore, the Iitaka dimension of the linear system corresponding to N_0' is submaximal.

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arXiv2021

Views on level $\mathit \ell$ curves, K3 surfaces and Fano threefolds

An analogue of the Mukai map $m_g: \mathcal P_g \to \mathcal M_g$ is studied for the moduli $\mathcal R_{g, \ell}$ of genus $g$ curves $C$ with a level $\ell$ structure. Let $\mathcal P^{\perp}_{g, \ell}$ be the moduli space of $4$-tuples $(S, \mathcal L, \mathcal E, C)$ so that $(S, \mathcal L)$ is a polarized K3 surface of genus $g$, $\mathcal E$ is orthogonal to $\mathcal L$ in Pic$S$ and defines a standard degree $\ell$ K3 cyclic cover of $S$, $C \in \vert \mathcal L \vert$. We say that $(S, \mathcal L, \mathcal E)$ is a level $\ell$ K3 surface. These exist for $\ell \leq 8$ and their families are known. We define a level $\ell$ Mukai map $r_{g, \ell}: \mathcal P^{\perp}_{g, \ell} \to \mathcal R_{g, \ell}$, induced by the assignment of $(S, \mathcal L, \mathcal E, C)$ to $ (C, \mathcal E \otimes \mathcal O_C)$. We investigate a curious possible analogy between $m_g$ and $r_{g, \ell}$, that is, the failure of the maximal rank of $r_{g, \ell}$ for $g = g_{\ell} \pm 1$, where $g_{\ell}$ is the value of $g$ such that $\dim \mathcal P^{\perp}_{g, \ell} = \dim \mathcal R_{g,\ell}$. This is proven here for $\ell = 3$. As a related open problem we discuss Fano threefolds whose hyperplane sections are level $\ell$ K3 surfaces and their classification.

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arXiv2020

The Igusa quartic and the Prym map, with some rational moduli

In this paper the ubiquity of the Igusa quartic $B \subset \mathbb P^4$ shows up again, this time related to the Prym map $\mathfrak p : \mathcal R_6 \to \mathcal A_5$. We introduce the moduli space $\mathcal X$ of those quartic threefolds $X$ cutting twice a quadratic section of $B$. A general $X$ is $30$-nodal and the intermediate Jacobian $J(X)$ of its natural desingularization is a $5$-dimensional p.p. abelian variety. Let $\frak j: \mathcal X \to \mathcal A_5$ be the period map sending $X$ to $J(X)$, in the paper we study $\frak j$ and its relation to $\frak p$. As is well known the degree of $\frak p$ is $27$ and its monodromy group endows any smooth fibre $F$ of $\frak p$ with the incidence configuration of $27$ lines of a cubic surface. Then the same monodromy defines a map $ \mathfrak j': \mathcal D_6 \to \mathcal A_5$ of degree $36$, with fibre the configuration of $36$ 'double-six' sets of lines of a cubic surface. We prove that $\frak j = \frak j' \circ ϕ$, where $ϕ: \mathcal X \to \mathcal D_6$ is birational. This provides a geometric description of $\frak j'$. Finally we describe the relations between the different moduli spaces considered and prove that some, including $\mathcal X$, are rational.

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arXiv2020

The unirationality of the moduli space of K3 surfaces of genus 22

Using the connection discovered by Hassett between the Noether-Lefschetz moduli space of special cubic fourfolds of discriminant 42 and the moduli space F_{22} of polarized K3 surfaces of genus 22, we show that the universal K3 surface over F_{22} is unirational.

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arXiv2019

On Morin configurations of higher length

This paper studies finite Morin configurations $F$ of planes in $\mathbb P^5$ having higher length. The uniqueness of the configuration of maximal cardinality $20$ is proven. This is related to the stable canonical genus $6$ curve $C_{\ell}$ union of the $10$ lines of a smooth quintic Del Pezzo surface $Y$ in $\mathbb P^5$ and to the Petersen graph. Families of length $\geq 16$, previously unknown, are constructed by smoothing partially $C_{\ell}$. A more general irreducible family of special configurations of length $\geq 11$, we name as Morin-Del Pezzo configurations, is considered and studied. This depends on $9$ moduli and is defined via the family of nodal and rational canonical curves of $Y$. The special relations between Morin-Del Pezzo configurations and the geometry of special threefolds, like the Igusa quartic or its dual Segre primal, are focused.

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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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