arXiv subjects
Publications and source records attributed to Gianfranco Casnati.
Let $X$ be a Fano threefold with index $i_X$ and fundamental line bundle $\mathcal O_X(h)$. We classify $μ$-semistable rank two bundles $\mathcal E$ on $X$ with $c_1(\mathcal E)=0$, $h^0(\mathcal E) \ne 0$ and $h^1(\mathcal E(-\lceil\frac{i_X}{2}\rceil h))=0$.
We study instanton and Ulrich bundles on hypersurfaces of the projective space, with a focus on special cubic fourfolds and generalized Pfaffians, notably defined by skew-symmetric endomorphisms of Steiner bundles. We prove that the acyclic extensions of instantons deform to Ulrich bundles and deduce that the existence of instantons of low rank and charge implies the existence of Ulrich bundles of low rank, which in turn forces the fourfold to lie in some Hassett divisor. Finally we take a closer look to divisors of cubics with discriminant 18 and 20.
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.
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.
In this very short note we give an elementary characteristic free proof of the result claimed in the title (see Theorem 1.2 for a more precise formulation), generalizing a recent result proved for Ulrich bundles over the complex field by V. Benedetti, P. Montero, Y. Prieto Montañez, S. Troncoso. Moreover, we also give a similar result about the twists of the cotangent bundle and make some comments about the possibility to obtain an analogous result for twists of the tangent bundle.
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.
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.
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.
Let $S$ be a regular surface endowed with a very ample line bundle $\mathcal O_S(h_S)$. Taking inspiration from a very recent result by D. Faenzi on $K3$ surfaces, we prove that if $\mathcal O_S(h_S)$ satisfies a short list of technical conditions, then such a polarized surface supports special Ulrich bundles of rank $2$. As applications, we deal with general embeddings of regular surfaces, pluricanonically embedded regular surfaces and some properly elliptic surfaces of low degree in $\mathbb P^N$. Finally, we also discuss about the size of the families of Ulrich bundles on $S$ and we inspect the existence of special Ulrich bundles on surfaces of low degree.
We give examples of surfaces which are Ulrich-wild, i.e. that support families of dimension $p$ of pairwise non-isomorphic, indecomposable, Ulrich bundles for arbitrary large $p$.
We deal with instanton bundles on the product ${\mathbb P}^1\times{\mathbb P}^2$ and the blow up of ${\mathbb P}^3$ along a line. We give an explicit construction leading to instanton bundles. Moreover, we also show that they correspond to smooth points of a unique irreducible component of their moduli space.
We propose a general definition of mathematical instanton bundle with given charge on any Fano threefold extending the classical definitions on $\mathbb P^3$ and on Fano threefold with cyclic Picard group. Then we deal with the case of the blow up of $\mathbb P^3$ at a point, giving an explicit construction of instanton bundles satisfying some important extra properties: moreover, we also show that they correspond to smooth points of a component of the moduli space.
Let $S$ be a surface with $p_g(S)=0$, $q(S)=1$ and endowed with a very ample line bundle $\mathcal O_S(h)$ such that $h^1\big(S,\mathcal O_S(h)\big)=0$. We show that such an $S$ supports families of dimension $p$ of pairwise non-isomorphic, indecomposable, Ulrich bundles for arbitrary large $p$. Moreover, we show that $S$ supports stable Ulrich bundles of rank $2$ if the genus of the general element in $\vert h\vert$ is at least $2$.
Let $S$ be a surface with $p_g(S)=q(S)=0$ and endowed with a very ample line bundle $\mathcal O_S(h)$ such that $h^1\big(S,\mathcal O_S(h)\big)=0$. We show that $S$ supports special (often stable) Ulrich bundles of rank $2$, extending a recent result by A. Beauville. Moreover, we show that such an $S$ supports families of dimension $p$ of pairwise non-isomorphic, indecomposable, Ulrich bundles for arbitrary large $p$ except for very few cases. We also show that the same is true for linearly normal non-special surface in $\mathbb P^4$ of degree at least $4$, Enriques surface and anticanonical rational surface.
Let $S$ be a geometrically ruled surface with invariant $e$ on a curve $C$. We deal with Ulrich line bundles and $μ$-stable special Ulrich bundles of rank $2$ on $S$ when $e\ge0$, slightly extending a recent result due to M. Aprodu, L. Costa and R.M. Miró-Roig. If $C$ is elliptic, we also prove that $S$ always supports Ulrich bundles of rank at most $2$, without any restriction on $e$. Finally, we show that in many cases $S$ supports families of dimension $p$ of pairwise non-isomorphic, indecomposable, Ulrich bundles for arbitrary large $p$.
Let $F\subseteq\mathbb P ^{a+1}$ be a non-degenerate $K3$ surface of degree $2a$, where $a\ge2$. In this paper we deal with Ulrich bundles on $F$ of rank $2$. We deal with their stability and we construct $K3$ surfaces endowed with families of non-special Ulrich bundles of rank $2$ for each $a\ge2$.
Let $F\subseteq\mathbb{P}^3$ be a smooth quartic surface and let $\mathcal{O}_F(h):=\mathcal{O}_{\mathbb{P}^3}(1)\otimes\mathcal{O}_F$. In the present paper we classify locally free sheaves $\mathcal{E}$ of rank $2$ on $F$ such that $c_1(\mathcal{E})=\mathcal{O}_F(2h)$, $c_2(\mathcal{E})=8$ and $h^1\big(F,\mathcal{E}(th)\big)=0$ for $t\in\mathbb{Z}$. We also deal with their stability.
Let $F\subseteq\mathbb{P}^3$ be a smooth determinantal quartic surface which is general in the Nöther-Lefschetz sense. In the present paper we give a complete classification of locally free sheaves $\mathcal{E}$ of rank $2$ on $F$ such that $h^1\big(F,\mathcal{E}(th)\big)=0$ for $t\in\mathbb{Z}$.