Related papers: Non-existence of low rank Ulrich bundles on Verone…

In this paper, we investigate the existence of Ulrich bundles on a smooth complete intersection of two $4$-dimensional quadrics in $\mathbb P^5$ by two completely different methods. First, we find good ACM curves and use Serre…

We use certain special prehomogeneous representations of algebraic groups in order to construct aCM vector bundles, possibly Ulrich, on certain families of hypersurfaces. Among other results, we show that a general cubic hypersurface of…

We show the existence of rank 6 Ulrich bundles on a smooth cubic fourfold. First, we construct a simple sheaf E of rank 6 as an elementary modification of an ACM bundle of rank 6 on a smooth cubic fourfold. Such an E appears as an extension…

We exhibit a class of extendable codimension $2$ subvarieties in a general hypersurface of dimension at least $4$ in projective space. As a consequence, we prove that a general hypersurface of degree $d$ and dimension at least $4$ does not…

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,…

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…

We characterize the vector bundles on G(1,4) that have no intermediate cohomology. We obtain them from extensions of the universal bundles and others related with them. In particular, we get a characterization of the universal vector…

We demonstrate the existence of a uniform and nonhomogeneous vector bundle $E$ of rank $(n-d)(m+1)-1$ over Grassmannian $\mathbb{G}(d,n)$, where $m>d$ and $1\le d \le n-d-1$ with a $\mathbb{P}$-homogeneity degree $h(E)=d$. Particularly, we…

We classify Ulrich vector bundles that are not big on smooth complex surfaces and threefolds.

In this paper, we give necessary and sufficient conditions for the existence of Ulrich bundles on cubic fourfold $X$ of given rank $r$. As consequences, we show that for every integer $r\ge 2$ there exists a family of indecomposable rank…

We study the resolution of an Ulrich bundle of arbitrary rank on the Segre fourfold $\PP^2\times\PP^2$. We characterize the Ulrich bundles $\Vv$ of arbitrary rank on $\PP^2\times\PP^2$ with $h^1(\Vv\otimes\Omega\boxtimes\Omega)=0$ or with…

In this paper, we consider the existence problem of Ulrich bundles on a rational homogeneous space $G/P$ of type $B$, $C$ or $D$. We show that if the Picard number of $G/P$ is greater than or equal to $2$, then there are no irreducible…

In this paper, we study equivariant vector bundles on partial flag varieties arising from Schur functors. We show that a partial flag variety with three or more steps does not admit an Ulrich bundle of this form with respect to the minimal…

We study the vector bundles without intermediate cohomology on Fano threefolds of index two, degree d=3,4,5 and Betti number one. We obtain a complete characterization in the case of rank-two vector bundles. For arbitrary rank, we give all…

I provide a construction of intrinsic weakly Ulrich bundles of large rank on any smooth complete surface in ${\bf P}^3$ over fields of characteristic $p>0$ and also for some classes of surfaces of general type in ${\bf P}^n$. I also…

The classification of algebraic vector bundles of rank 2 over smooth affine fourfolds is a notoriously difficult problem. Isomorphism classes of such vector bundles are not uniquely determined by their Chern classes, in contrast to the…

The existence of stable ACM vector bundles of high rank on algebraic varieties is a challenging problem. In this paper, we study stable Ulrich bundles (that is, stable ACM bundles whose corresponding module has the maximum number of…

We study the positivity of the first Chern class of a rank r Ulrich vector bundle E on a smooth n-dimensional variety $X \subseteq \mathbb P^N$. We prove that $c_1(E)$ is very positive on every subvariety not contained in the union of lines…

In this article, we prove that any smooth projective variety $X$ which is a double cover of the projective space $\mathbb{P}^n$ ($n\geq 2$) admits an Ulrich bundle. When $n=2$, we show that on any such $X$, there is an Ulrich bundle of rank…

This paper investigates property QR(3) for Veronese embeddings over an algebraically closed field of characteristic $3$. We determine the rank index of $(\mathbb{P}^n , \mathcal{O}_{\mathbb{P}^n} (d))$ for all $n \geq 2$, $d \geq 3$,…