Searcharxiv⌕ Search

arXiv subjects

Rita Pardini

Publications and source records attributed to Rita Pardini.

At least 19 recordsLinked to original sources
arXiv2026

The parity of theta characteristics is preserved by infinitesimal deformations

In this note, given a family of relative dimension one over a smooth curve, we determine the parity of the restriction of a relative theta characteristic to an arbitrary multiple of a fiber in terms of the parity of the restriction to a general fibre. This result can be regarded as a variant of the well-known theorem on the invariance of the parity of theta characteristics in families. As a corollary, we obtain that the torsion subsheaf of the first higher direct image sheaf of a relative theta characteristic splits as a direct sum of two isomorphic sheaves.

math.AG↗
arXiv2026

Surfaces with canonical map of odd degree

Let $S$ be a smooth complex minimal surface of general type with $p_g:=h^0(K_S)\ge 4$ whose canonical map is generically finite of odd degree $d>1$ onto a surface $Σ$. We assume that the general canonical curve of $S$ is smooth and that $Σ$ is ruled by lines, and we prove: - $p_g\le d+2$ - $Σ$ is a cone over the rational normal curve of degree $p_g-2$ in ${\mathbb P}^{p_g-1}$ - $p_g=d+2$ can occur only for $d=3,9,11$. As a byproduct, we refine previous results by Beauville and Xiao by proving that if one drops the assumption that $Σ$ is ruled by lines then $d\le 5$ if $p_g\ge 112$. The case $d=3$ being completely classified by the first two named authors, we focus on $d=5$, showing that $p_g\le 5$ and that for $p_g=5$ the surface $S$ has a pencil $|C|$ with $C^2=1$ and $K_SC=5$. These results suggest that the answer to the question whether the surfaces with canonical map of odd degree $d>1$ have bounded invariants could be positive, in sharp contrast with the case of even degree.

math.AG↗
arXiv2026

Numerical inequalities for quasi-projective surfaces

Let $V$ be a smooth quasi-projective complex surface with compactification $(X,D)$ and set $\overline P_1(V):=h^0(X,K_X+D)$, $\overline q(V):=h^0(X,Ω^1_X(\log D))$. We prove that $\overline P_1(V)\ge \overline q(V)-1$ if $V$ has maximal Albanese dimension and $\overline P_1(V)\ge\frac 16( \overline q(V)-5)$ otherwise. Both bounds are sharp.

math.AG↗
arXiv2025

On some boundary divisors in the moduli spaces of stable Horikawa surfaces with $K^2=2p_g-3$

We describe the normal stable surfaces with K^2=2p_g-3 and p_g>14 whose only non canonical singularity is a cyclic quotient singularity of type 1/4k(1,2k-1) and the corresponding locus D inside the KSBA moduli space of stable surfaces. More precisely, we show that: (1) a general point of any irreducible component of D corresponds to a surface with a singularity of type 1/4(1,1), (2) the closure of D is a divisor contained in the closure of the Gieseker moduli space of canonical models of surfaces with K^2=2p_g-3 and intersects all the components of such closure, and (3) the KSBA moduli space is smooth at a general point of D. In addition, we show that D has 1 or 2 irreducible components, depending on the residue class of p_g modulo 4.

math.AG↗
arXiv2024

2-Gorenstein stable surfaces with $K_X^2 = 1$ and $χ(X) = 3$

The compactification $\overline M_{1,3}$ of the Gieseker moduli space of surfaces of general type with $K_X^2 =1 $ and $χ(X)=3$ in the moduli space of stable surfaces parametrises so-called stable I-surfaces. We classify all such surfaces which are 2-Gorenstein into four types using a mix of algebraic and geometric techniques. We find a new divisor in the closure of the Gieseker component and a new irreducible component of the moduli space.

math.AG↗
arXiv2024

A footnote to a theorem of Kawamata

Kawamata has shown that the quasi-Albanese map of a quasi-projective variety with log-irregularity equal to the dimension and log-Kodaira dimension 0 is birational. In this note we show that under these hypotheses the quasi-Albanese map is proper in codimension 1 as conjectured by Iitaka.

math.AG↗
arXiv2024

On the Hopf Problem and a Conjecture of Liu-Maxim-Wang

We discuss an approach towards the Hopf problem for aspherical smooth projective varieties recently proposed by Liu, Maxim, and Wang in [LMW21]. In complex dimension two, we point out that this circle of ideas suggests an intriguing conjecture regarding the geography of aspherical surfaces of general type.

math.AG↗
arXiv2023

On the existence of ramified abelian covers

Given a normal complete variety $Y$ over an algebraically closed field $\mathbb K$, distinct effective Weil divisors $D_1,... D_n$ of $Y$ and positive integers $d_1,... d_n$, we spell out the conditions for the existence of an abelian cover of $Y$ branched with order $d_i$ on $D_i$. As an application, we prove that a cover of a normal complete toric variety branched on the torus-invariant divisors is itself a toric variety if the characteristic of $\mathbb K$ is equal to 0 or if the cover is Galois of degree not divisible by the characteristic.

math.AG↗
arXiv2022

Effective characterization of quasi-abelian surfaces

Let V be a smooth quasi-projective complex surface such that the three first logarithmic plurigenera are equal to 1 and the logarithmic irregularity is equal to 2. We prove that the quasi-Albanese morphism of V is birational and there exists a finite set S such that the quasi-Albanese map is proper over the complement of S in the quasi-Albanese variety A(V) of V. This is a sharp effective version of a classical result of Iitaka.

math.AG↗
arXiv2022

Degeneration of Hodge structures on I-surfaces

Work of Green, Griffiths, Laza, and Robles suggests that the moduli space of (smoothable) stable surfaces should admit a natural stratification defined via Hodge theoretic data. In the case of stable surfaces with $K_X^2 = 1$ and $χ(X) = 3$ we compute the Hodge type of all examples known to us and show that all predicted degenerations are geometrically realised.

math.AG↗
arXiv2022

Numerical properties of exceptional divisors of birational morphisms of smooth surfaces

We make a very detailed analysis of the numerical properties of effective divisors whose support is contained in the exceptional locus of a birational morphism of smooth projective surfaces. As an application we extend Miyaoka's inequality on the number of canonical singularities on a projective normal surface with non-negative Kodaira dimension to the non minimal case, obtaining a slightly better result than known extensions by Megyesi and Langer.

math.AG↗
arXiv2021

On T-divisors and intersections in $\overline{M}_{1,3}$

The moduli space of stable surfaces with $K_X^2 = 1$ and $χ(X) = 3$ has at least two irreducible components that contain surfaces with T-singularities. We show that the two known components intersect transversally in a divisor. Moreover, we exhibit other new boundary divisors and study how they intersect one another.

math.AG↗
arXiv2021

Deformations of semi-smooth varieties

For a singular variety X, an essential step to determine its smoothability and study its deformations is the understanding of the tangent sheaf and of the sheaf T^1_X:=ext^1(Omega_X,O_X). A variety is semi-smooth if its singularities are étale locally the product of a double crossing point (uv=0) or a pinch point (u^2-v^2w=0) with affine space; equivalently, if it can be obtained by gluing a smooth variety along a smooth divisor via an involution with smooth quotient. Our main result is the explicit computation of the tangent sheaf and the sheaf T^1_X for a semi-smooth variety X in terms of the gluing data.

math.AG↗
arXiv2021

Smoothing semi-smooth Stable Godeaux surfaces

We show that all the semi-smooth stable complex Godeaux surfaces, classified in [FPR18a], are smoothable, and that the moduli stack is smooth of the expected dimension 8 at the corresponding points.

math.AG↗
arXiv2021

On the degree of the canonical map of a surface of general type

Let X be a minimal complex surface of general type such that its image via the canonical map is a surface; we denote by d the degree of the canonical map. In this expository work, first of all we recall the known possibilities for the canonical image and for d when the canonical map is not birational, which are quite a few, and then we consider the question of producing concrete examples for all of them. We present the two main methods of construction of such examples and we give several instances of their application. We end the paper outlining the state of the art on this topic and raising several questions.

math.AG↗