arXiv subjects

Vincenzo Antonelli

Publications and source records attributed to Vincenzo Antonelli.

10 recordsLinked to original sources

arXiv2026

Stratification of moduli spaces of instantons on the Segre product of three lines via 't Hooft bundles

Let $X$ be the Segre product of three projective lines. For a fixed effective divisor $D$ on $X$, we introduce the notions of $D$-'t Hooft, $(D_i,D_j)$-special and $D$-sectional special bundle. The varieties parameterizing these bundles yield a natural stratification of the moduli space of stable instanton bundles with fixed Chern classes. After characterizing the curves associated with these bundles via Serre correspondence, we describe the corresponding Hilbert schemes. Using this description, we analyze the moduli spaces of $h_i$-'t Hooft bundles and the smaller strata of $(h_i,h_j)$-special and $(h_i)$-sectional special bundles. Finally, we provide a detailed study of the low-charge cases.

math.AG↗

arXiv2025

't Hooft bundles on the complete flag threefold and moduli spaces of instantons

In this work we study the moduli spaces of instanton bundles on the flag twistor space $F:=F(0,1,2)$. We stratify them in terms of the minimal twist supporting global sections and we introduce the notion of (special) 't Hooft bundle on $F$. In particular we prove that there exist $μ$-stable 't Hooft bundles for each admissible charge $k$. We completely describe the geometric structure of the moduli space of (special) 't Hooft bundles for arbitrary charge $k$. Along the way to reach these goals, we describe the possible structures of multiple curves supported on some rational curves in $F$ as well as the family of del Pezzo surfaces realized as hyperplane sections of $F$. Finally we investigate the splitting behaviour of 't Hooft bundles when restricted to conics.

math.AG↗

arXiv2024

Steiner representations of hypersurfaces

Let $X\subseteq{\mathbb P}^{n+1}$ be an integral hypersurface of degree $d$. We show that each locally Cohen-Macaulay instanton sheaf $\mathcal E$ on $X$ with respect to $\mathcal O_X\otimes\mathcal O_{\mathbb P^{n+1}}(1)$ in the sense of Definition 1.3 in arXiv:2205.04767 [math.AG] yields the existence of Steiner bundles $\mathcal G$ and $\mathcal F$ on $\mathbb P^{n+1}$ of the same rank $r$ and a morphism $φ\colon \mathcal G(-1)\to\mathcal F^\vee$ such that the form defining $X$ to the power $\mathrm{rk}(\mathcal E)$ is exactly $\det(φ)$. We inspect several examples for low values of $d$, $n$ and $\mathrm{rk}(\mathcal E)$. In particular, we show that the form defining a smooth integral surface in $\mathbb P^3$ is the pfaffian of some skew-symmetric morphism $φ\colon \mathcal F(-1)\to\mathcal F^\vee$, where $\mathcal F$ is a suitable Steiner bundle on $\mathbb P^3$ of sufficiently large even rank.

math.AG↗

arXiv2024

On varieties with Ulrich twisted conormal bundles

We study varieties $X \subset P^r$ such that is $N_X^*(k)$ is an Ulrich vector bundle for some integer $k$. We first prove that such an $X$ must be a curve. Then we give several examples of curves with $N_X^*(k)$ an Ulrich vector bundle.

math.AG↗

arXiv2022

Instanton sheaves on projective schemes

A $h$-instanton sheaf on a closed subscheme $X$ of some projective space endowed with an ample and globally generated line bundle $\mathcal{O}_X(h)$ is a coherent sheaf whose cohomology table has a certain prescribed shape. In this paper we deal with $h$-instanton sheaves relating them to Ulrich sheaves. Moreover, we study $h$-instanton sheaves on smooth curves and surfaces, cyclic $n$-folds, Fano $3$-folds and scrolls over arbitrary smooth curves. We also deal with a family of monads associated to $h$-instanton bundles on varieties satisfying some mild extra technical conditions.

math.AG↗

arXiv2021

Even and odd instanton bundles on Fano threefolds

We define non-ordinary instanton bundles on Fano threefolds $X$ extending the notion of (ordinary) instanton bundles. We determine a lower bound for the quantum number of a non-ordinary instanton bundle, i.e. the degree of its second Chern class, showing the existence of such bundles for each admissible value of the quantum number when $i_X\ge 2$ or $i_X=1$, $\mathrm{Pic}(X)$ is cyclic and $X$ is ordinary. In these cases we deal with the component inside the moduli spaces of simple bundles containing the vector bundles we construct and we study their restriction to lines. Finally we give a monadic description of non-ordinary instanton bundles on $\mathbb{P}^3$ and the smooth quadric studying their loci of jumping lines, when of the expected codimension.

math.AG↗

arXiv2021

Instanton bundles on $\mathbb{P}^1\times\mathbb{F}_1$

In this paper we deal with a particular class of rank two vector bundles (\emph{instanton} bundles) on the Fano threefold of index one $F:=\mathbb{F}_1 \times \mathbb{P}^1$. We show that every instanton bundle on $F$ can be described as the cohomology of a monad whose terms are free sheaves. Furthermore we prove the existence of instanton bundles for any admissible second Chern class and we construct a nice component of the moduli space where they sit. Finally we show that minimal instanton bundles (i.e. with the least possible degree of the second Chern class) are aCM and we describe their moduli space.

math.AG↗

arXiv2020

H-instanton bundles on three-dimensional polarized projective varieties

We propose a notion of instanton bundle (called $H$-instanton bundle) on any projective variety of dimension three polarized by a very ample divisor $H$, that naturally generalizes the ones on $\mathbb{P}^3$ and on the flag threefold $F(0,1,2)$. We discuss the cases of Veronese and Fano threefolds. Then we deal with $H$-instanton bundles $\mathcal{E}$ on three-dimensional rational normal scrolls $S(a_0,a_1,a_2)$. We give a monadic description of $H$-instanton bundles and we prove the existence of $μ$-stable $H$-instanton bundles on $S(a_0,a_1,a_2)$ for any admissible charge $k=c_2(\mathcal{E})H$. Then we deal in more detail with $S(a,a,b)$ and $S(a_0,a_1,a_2)$ with $a_0+a_1>a_2$ and even degree. Finally we describe a nice component of the moduli space of $μ$-stable bundles whose points represent $H$-instantons.

math.AG↗

arXiv2020

Characterization of Ulrich bundles on Hirzebruch surfaces

In this work we characterize Ulrich bundles of any rank on polarized rational ruled surfaces over $\mathbb{P}^1$. We show that every Ulrich bundle admits a resolution in terms of line bundles. Conversely, given an injective map between suitable totally decomposed vector bundles, we show that its cokernel is Ulrich if it satisfies a vanishing in cohomology. As a consequence we obtain, once we fix a polarization, the existence of Ulrich bundles for any admissible rank and first Chern class. Moreover we show the existence of stable Ulrich bundles for certain pairs $(\textrm{rk}(E),c_1(E))$ and with respect to a family of polarizations. Finally we construct examples of indecomposable Ulrich bundles for several different polarizations and ranks.

math.AG↗

arXiv2019

Instanton bundles on the Segre threefold with Picard number three

We study instanton bundles $E$ on $\mathbb{P}^1\times \mathbb{P}^1 \times \mathbb{P}^1$. We construct two different monads which are the analog of the monads for instanton bundles on $\mathbb P^3$ and on the flag threefold $F(0,1,2)$. We characterize the Gieseker semistable cases and we prove the existence of $μ$-stable instanton bundles generically trivial on the lines for any possible $c_2(E)$. We also study the locus of jumping lines.

math.AG↗