Let $E$ be a slope-semistable vector bundle on smooth projective curve $C$. Assume, $ \pi : G= Gr_{C}(r, E) \to C$ denote the relative Grassmannian of rank $r$ quotients of $E$. We prove that for every very ample polarization $L= O_{G}(1) \otimes \pi^{*} A$ the polarized variety $(G,L)$ possess an Ulrich bundle. The construction globalizes homogeneous ulrich bundle on Grassmannian fibres. The remaining cohomology vanishing problem reduced to an acyclicity criterion on the base curve. We finally use Falting's cohomological charaterization of semistability.
Let $G=\operatorname{Gr}(k,V)$ be a complex Grassmannian in its Pl$\"u$cker embedding, and let $Y_E$ be a general codimension-$r$ linear section. We study the open Hilbert scheme $R_2(Y_E)$ of smooth conics using the kernel and span of a conic. These define a flag variety $B=\operatorname{Fl}(k-2,k+2;V)$ and a relative...
Suppose we are given an arbitrary line bundle $\mathcal{L}$ on a complex algebraic variety $X$, not necessarily smooth. For a positive integer $N$, suppose there exists a global section $s \in \Gamma(X, \mathcal{L}^{N})$ that defines an effective Cartier divisor $D$. If we denote $\pi: Y \rightarrow X$ to be the $N$-fo...
Let $E$ be a vector bundle on a normal complex projective variety. We prove that slope stability of $\mathop{Sym}^2E$ and Zariski density of the reductions of $E$ modulo $p$ that are not strongly geometrically stable imply that the bundle $E$ is projectively unitary flat. This generalizes Shepherd--Barron's result from...
Answering a 1985 question of Moret-Bailly, we prove by way of counterexample that if a line bundle $\mathscr{L}$ on a degeneration of abelian varieties $G \to \mathrm{Spec}(R)$ is not symmetric, then $\mathscr{L}$ being ample on $G$ is not necessarily equivalent to $\mathscr{L}$ being ample on the generic fiber of $G$....
Let $G=\mathrm{SL}_n(\mathbf C)$ and let $P\subset G$ be a standard parabolic subgroup with Levi factor $L$. For a $G$-dominant weight $\lambda$, consider the vector bundle on $T^*(G/P)$ obtained by pulling back the vector bundle on $G/P$ associated to the irreducible $L$-module $V_L(\lambda)^*$. We express its cohomol...
For a vector bundle $E$ on a smooth projective curve $C$, one defines the Quot scheme $Q:=Q_d(E,C)$ parametrizing subsheaves of $E$ having length $d$ torsion quotients. We describe the indecomposable components of the tangent bundle $T_Q$ of $Q$, thus characterizing when $T_Q$ is indecomposable. We also characterize wh...
