Searcharxiv⌕ Search

arXiv subjects

Davide Frapporti

Publications and source records attributed to Davide Frapporti.

16 recordsLinked to original sources
arXiv2026

Cohomologically or numerically trivial automorphisms of surfaces of general type

Our main result is the determination of the respective groups $ Aut_\mathbb{Z}(S) $ of cohomologically trivial automorphisms and $ Aut_\mathbb{Q}(S) $ of numerically trivial automorphisms for the reducible fake quadrics, that is, the surfaces $S$ isogenous to a product with $q=p_g=0$. In this way we produce new record winning examples: a surface $S$ with $|Aut_\mathbb{Q}(S)| =192$, and a surface whose cohomology has torsion with nontrivial $ Aut_\mathbb{Z}(S) \cong \mathbb{Z}/2.$

math.AG↗
arXiv2025

Pluricanonical Geometry of Varieties Isogenous to a Product and Abelian Covers

We study canonical and pluricanonical maps of varieties isogenous to a product of curves, i.e., quotients of the form $X = (C_1 \times \dots \times C_n)/G$ with $g(C_i)\ge 2$ and $G$ acting freely. For this purpose, we provide a technical result which is of general interest: a decomposition theorem for pluricanonical systems of abelian covers. This theorem provides an effective tool for the explicit study of geometric properties, such as base loci and the birationality of pluricanonical maps. For threefolds isogenous to a product, we prove that the 4-canonical map is birational for $p_g \ge 5$ and construct an example attaining the maximal canonical degree for this class of threefolds. In this example, the canonical map is the normalization of its image, which admits isolated non-normal singularities. Computational classifications also reveal threefolds where the bicanonical map fails to be birational, even in the absence of genus-2 fibrations. This illustrates an interesting phenomenon similar to the non-standard case for surfaces.

math.AG↗
arXiv2024

On the cohomologically trivial automorphisms of elliptic surfaces I: $\chi(S)=0$

In this first part we describe the group $Aut_{\mathbb{Z}}(S)$ of cohomologically trivial automorphisms of a properly elliptic surface (a minimal surface $S$ with Kodaira dimension $\kappa(S)=1$), in the initial case $ \chi(\mathcal{O}_S) =0$. In particular, in the case where $Aut_{\mathbb{Z}}(S)$ is finite, we give the upper bound 4 for its cardinality, showing more precisely that if $Aut_{\mathbb{Z}}(S)$ is nontrivial, it is one of the following groups: $\mathbb{Z}/2, \mathbb{Z}/3, (\mathbb{Z}/2)^2$. We also show with easy examples that the groups $\mathbb{Z}/2, \mathbb{Z}/3$ do effectively occur. Respectively, in the case where $Aut_{\mathbb{Z}}(S)$ is infinite, we give the sharp upper bound 2 for the number of its connected components.

math.AG↗
arXiv2022

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.

math.AG↗
arXiv2020

On the minimal model of semi-isogenous mixed surfaces

The aim of this paper is to determine minimal models of the semi-isogenous mixed surfaces with $χ=1$ and $K^2>0$ constructed by Cancian and Frapporti. In order to do this, we further develop the idea of orbit divisors introduced by Frapporti and Lee, to construct effective divisors on surfaces isogenous to a product of mixed type, extending it to the semi-isogenous mixed surfaces.

math.AG↗
arXiv2017

On Semi-isogenous mixed surfaces

Let $C$ be a smooth projective curve and $G$ a finite subgroup of $\mathrm{Aut}(C)^2\rtimes \mathbb Z_2$ whose action is \textit{mixed}, i.e.~there are elements in $G$ exchanging the two isotrivial fibrations of $C\times C$. Let $G^0\triangleleft G$ be the index two subgroup $G\cap\mathrm{Aut}(C)^2$. If $G^0$ acts freely, then $X:=(C\times C)/G$ is smooth and we call it \textit{semi-isogenous mixed surface}. In this paper we give an algorithm to determine semi-isogenous mixed surfaces with given geometric genus, irregularity and self-intersection of the canonical class. As an application we classify irregular semi-isogenous mixed surfaces with $K^2>0$ and geometric genus equal to the irregularity; the regular case is subjected to some computational restrictions. In this way we construct new examples of surfaces of general type with $χ=1$. We provide an example of a minimal surface of general type with $K^2=7$ and $p_g=q=2$.

math.AG↗
arXiv2017

On Threefolds Isogenous to a Product of Curves

A threefold isogenous to a product of curves $X$ is a quotient of a product of three compact Riemann surfaces of genus at least two by the free action of a finite group. In this paper we study these threefolds under the assumption that the group acts diagonally on the product. We show that the classification of these threefolds is a finite problem, present an algorithm to classify them for a fixed value of $χ(\mathcal O_X)$ and explain a method to determine their Hodge numbers. Running an implementation of the algorithm we achieve the full classification of threefolds isogenous to a product of curves with $χ(\mathcal O_X)=-1$, under the assumption that the group acts faithfully on each factor.

math.AG↗
arXiv2015

Generalized Burniat type surfaces and Bagnera-de Franchis varieties

In this article we construct three new families of surfaces of general type with p_g = q = 0,K^2 = 6, and seven new families of surfaces of general type with p_g = q = 1, K^2 = 6, realizing 10 new fundamental groups. We also show that these families correspond to pairwise distinct irreducible connected components of the Gieseker moduli space of surfaces of general type. We achieve this using two different main ingredients. First we introduce a new class of surfaces, called generalized Burniat type surfaces, and we completely classify them (and the connected com- ponents of the moduli space containing them). Second, we introduce the notion of Bagnera-de Franchis varieties: these are the free quotients of an Abelian variety by a cyclic group (not consisting only of translations). For these we develop some basic results.

math.AG↗
arXiv2014

Bloch's conjecture for Generalized Burniat Type surfaces with $p_g=0$

The aim of this article is to prove Bloch's conjecture, asserting that the group of rational equivalence classes of zero cycles of degree 0 is trivial for surfaces with geometric genus zero, for regular generalized Burniat type surfaces. The technique is the method of "enough automorphisms" introduced by Inose-Mizukami in a simplified version due to the first author.

math.AG↗
arXiv2014

The fundamental group and torsion group of Beauville surfaces

We give a survey on the fundamental group of surfaces isogenous to a higher product. If the surfaces are regular, e.g. if they are Beauville surfaces, the first homology group is a finite group. We present a MAGMA script which calculates the first homology groups of regular surfaces isogenous to a product.

math.AG↗
arXiv2013

Mixed quasi-étale quotients with arbitrary singularities

A mixed quasi-étale quotient is the quotient of the product of a curve of genus at least 2 with itself by the action of a group which exchanges the two factors and acts freely out of a finite subset. A mixed quasi-étale surface is the minimal resolution of its singularities. We produce an algorithm computing all mixed quasi-étale surfaces with given geometric genus, irregularity, and self-intersection of the canonical class. We prove that all irregular mixed quasi-étale surfaces of general type are minimal. As application, we classify all irregular mixed quasi étale surfaces of general type with genus equal to the irregularity, and all the regular ones with K^2>0, thus constructing new examples of surfaces of general type with χ=1. We mention the first example of a minimal surface of general type with p_g=q=1 and Albanese fibre of genus bigger than K^2.

math.AG↗
arXiv2013

Mixed quasi-étale surfaces, new surfaces of general type with $p_g=0$ and their fundamental group

We call a projective surface $X$ mixed quasi-étale quotient if there exists a curve $C$ of genus $g(C)\geq 2$ and a finite group $G$ that acts on $C\times C$ exchanging the factors such that $X=(C\times C)/G$ and the map $C\times C \rightarrow X$ has finite branch locus. The minimal resolution of its singularities is called mixed quasi-étale surface. We study the mixed quasi-étale surfaces under the assumption that $(C\times C)/G^0$ has only nodes as singularities, where $G^0\triangleleft G$ is the index two subgroup of the elements that do not exchange the factors. We classify the minimal regular surfaces with $p_g=0$ whose canonical model is a mixed quasi-étale quotient as above. All these surfaces are of general type and as an important byproduct, we provide an example of a numerical Campedelli surface with topological fundamental group $\bbZ_4$, and we realize 2 new topological types of surfaces of general type. Three of the families we construct are $\bbQ$-homology projective planes.

math.AG↗
arXiv2009

On the non-existence of orthogonal instanton bundles on P^(2N+1)

In this paper we prove that there do not exist orthogonal instanton bundles on P^(2n+1) . In order to demonstrate this fact, we propose a new way of representing the invariant, introduced by L. Costa and G. Ottaviani, related to a rank 2n instanton bundle on P^(2n+1) .

math.AG↗