Vector bundles on flag varieties

We study vector bundles on flag varieties over an algebraically closed field k. In the first part, we suppose $G=G_k(d,n)$ $(2\le d\le n-d)$ to be the Grassmannian parameterizing linear subspaces of dimension d in $k^n$, where k is an algebraically closed field of characteristic $p>0$. Let E be a uniform vector bundle over G of rank $r\le d$. We show that E is either a direct sum of line bundles or a twist of the pullback of the universal subbundle $H_d$ or its dual $H_d^{\vee }$ by a series of absolute Frobenius maps. In the second part, splitting properties of vector bundles on general flag varieties $F(d_1,\text{…},d_s)$ in characteristic zero are considered. We prove a structure theorem for bundles over flag varieties which are uniform with respect to the ith component of the manifold of lines in $F(d_1,\text{…},d_s)$. Furthermore, we generalize the Grauert–M$\ddot{\mathrm{u}}$lich–Barth theorem to flag varieties. As a corollary, we show that any strongly uniform i-semistable $(1\le i\le n-1)$ bundle over the complete flag variety splits as a direct sum of special line bundles.     

On the equivalence of certain quasi-Hermitian varieties

By Aguglia et al., new quasi-Hermitian varieties ${\mathrm{ℳ}}_{\alpha ,\beta }$ in $\text{PG}\left(r,q^2\right)$ depending on a pair of parameters $\alpha ,\beta$ from the underlying field $\text{GF}\left(q^2\right)$ have been constructed. In the present paper we study the structure of the lines contained in ${\mathrm{ℳ}}_{\alpha ,\beta }$ and consequently determine the projective equivalence classes of such varieties for $q$ odd and $r=3$. As a byproduct, we also prove that the collinearity graph of ${\mathrm{ℳ}}_{\alpha ,\beta }$ is connected with diameter 3 for $q\equiv 1\left(\mathrm{mod}4\right)$.     

Rank gain of Jacobians over number field extensions with prescribed Galois groups

We investigate the rank gain of elliptic curves, and more generally, Jacobian varieties, over non-Galois extensions whose Galois closure has a Galois group permutation-isomorphic to a prescribed group G (in short, “G-extensions”). In particular, for alternating groups and (an infinite family of) projective linear groups G, we show that most elliptic curves over (for example) $\mathbb{Q}$ gain rank over infinitely many G-extensions, conditional only on the parity conjecture. More generally, we provide a theoretical criterion, which allows to deduce that “many” elliptic curves gain rank over infinitely many G-extensions, conditional on the parity conjecture and on the existence of geometric Galois realizations with group G and certain local properties.