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Davide Cesare Veniani

Publications and source records attributed to Davide Cesare Veniani.

18 recordsLinked to original sources
arXiv2025

The Enriques surface of minimal entropy

Lehmer's number $λ_{10}$ is the smallest dynamical degree greater than $1$ that can occur for an automorphism of an algebraic surface. We show that $λ_{10}$ cannot be realized by automorphisms of Enriques surfaces in odd characteristic, extending a result of Oguiso over the complex numbers. In contrast, we prove that in characteristic $2$ there exists a unique Enriques surface that admits an automorphism with dynamical degree $λ_{10}$. We also provide explicit equations for the surface as well as for all conjugacy classes of automorphisms that realize $λ_{10}$.

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arXiv2025

Idoneal genera and K3 surfaces covering an Enriques surface

Idoneal genera are a generalization of Euler's idoneal numbers. We enumerate all idoneal genera by means of the Smith--Minkowski--Siegel mass formula. As an application, we classify transcendental lattices of K3 surfaces covering an Enriques surface.

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arXiv2024

Symplectic rigidity of O'Grady's tenfolds

We prove that any symplectic automorphism of finite order of an irreducible holomorphic symplectic manifold of O'Grady's 10-dimensional deformation type is trivial.

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arXiv2023

Hensel lifting algorithms for quadratic forms

We provide an algorithm to compute generators of the orthogonal group of the discriminant group associated to an integral quadratic lattice over the integers. We give a closed formula for its order.

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arXiv2022

Enriques surfaces of non-degeneracy 3

We classify all non-extendable 3-sequences of half-fibers on Enriques surfaces. If the characteristic is different from 2, we prove in particular that every Enriques surface admits a 4-sequence, which implies that every Enriques surface is the minimal desingularization of an Enriques sextic, and that every Enriques surface is birational to a Castelnuovo quintic.

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arXiv2021

Symplectic birational transformations of finite order on O'Grady's sixfolds

We prove that any symplectic automorphism of finite order on a manifold of type OG6 acts trivially on the Beauville--Bogomolov--Fujiki lattice and that any birational transformation of finite order acts trivially on its discriminant group. Moreover, we classify all possible invariant and coinvariant sublattices.

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arXiv2021

Enriques involutions on pencils of K3 surfaces

The three pencils of K3 surfaces of minimal discriminant whose general element covers at least one Enriques surface are Kondō's pencils I and II, and the Apéry--Fermi pencil. We enumerate and investigate all Enriques surfaces covered by their general elements.

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arXiv2021

Counting elliptic fibrations on K3 surfaces

We solve the problem of counting jacobian elliptic fibrations on an arbitrary complex projective K3 surface up to automorphisms. We then illustrate our method with several explicit examples.

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arXiv2020

Lines on K3 quartic surfaces in characteristic 3

We investigate the number of straight lines contained in a K3 quartic surface \(X\) defined over an algebraically closed field of characteristic 3. We prove that if \(X\) contains 112 lines, then \(X\) is projectively equivalent to the Fermat quartic surface; otherwise, \(X\) contains at most 67 lines. We improve this bound to 58 if \(X\) contains a star (ie four distinct lines intersecting at a smooth point of \(X\)). Explicit equations of three 1-dimensional families of smooth quartic surfaces with 58 lines, and of a quartic surface with 8 singular points and 48 lines are provided.

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arXiv2020

Semi-Calabi-Yau orbifolds and mirror pairs

We generalize the cohomological mirror duality of Borcea and Voisin in any dimension and for any number of factors. Our proof applies to all examples which can be constructed through Berglund-Hübsch duality. Our method is a variant of the so-called Landau-Ginzburg/Calabi-Yau correspondence of Calabi-Yau orbifolds with an involution that does not preserve the volume form. We deduce a version of mirror duality for the fixed loci of the involution, which are beyond the Calabi-Yau category and feature hypersurfaces of general type.

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arXiv2019

Enriques involutions on singular K3 surfaces of small discriminants

We classify Enriques involutions on a K3 surface, up to conjugation in the automorphism group, in terms of lattice theory. We enumerate such involutions on singular K3 surfaces with transcendental lattice of discriminant smaller than or equal to 36. For 11 of these K3 surfaces, we apply Borcherds method to compute the automorphism group of the Enriques surfaces covered by them. In particular, we investigate the structure of the two most algebraic Enriques surfaces.

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arXiv2017

Symmetries and equations of smooth quartic surfaces with many lines

We provide explicit equations of some smooth complex quartic surfaces with many lines, including all 10 quartics with more than 52 lines. We study the relation between linear automorphisms and some configurations of lines such as twin lines and special lines. We answer a question by Oguiso on a determinantal presentation of the Fermat quartic surface.

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arXiv2016

Lines on K3 quartic surfaces in characteristic 2

We prove that a K3 quartic surface defined over a field of characteristic 2 can contain at most 68 lines. If it contains 68 lines, then it is projectively equivalent to a member of a 1-dimensional family found by Rams and Schütt.

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arXiv2015

On prime degree isogenies between K3 surfaces

We classify prime order isogenies between algebraic K3 surfaces whose rational transcendental Hodges structures are not isometric. The morphisms of Hodge structures induced by these isogenies are correspondences by algebraic classes on the product fourfolds; however, they do not satisfy the hypothesis of the well-known Mukai--Nikulin theorem. As an application we describe isogenies obtained from K3 surfaces with an action of a symplectic automorphism of prime order.

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arXiv2015

The maximum number of lines lying on a K3 quartic surface

We show that there cannot be more than 64 lines on a quartic surface admitting isolated rational double points over an algebraically closed field of characteristic $p \neq 2,\,3$, thus extending Segre--Rams--Schütt theorem. Our proof offers a deeper insight into the triangle-free case and takes advantage of a special configuration of lines, thereby avoiding the technique of the flecnodal divisor. We provide several examples of non-smooth K3 quartic surfaces with many lines.

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