THE RESTRICTED TANGENT
BUNDLE OF SMOOTH CURVES
IN GRASSMANNIANS
AND CURVES IN FLAG VARIETIES

EDOARDO BALLICO AND LUCIANA RAMELLA

Abstract:

Let $X$ be a smooth curve of genus $g\ge2$ over an algebraically closed base field $k$ of any characteristic. Denote by $G(r,\nu)$ the Grassmannian of the rank $r$ quotients of $k^\nu$ and by ${\cal Q}$ the universal quotient bundle of $G(r,\nu)$. Let us consider degree $d$ embeddings $\varphi:X\to G(r,\nu)$. We prove that, for $d\ge\nu+r(g-1)$ and $(\nu,r,d)\ne(4,2,2g+2)$, varying $\varphi$ we obtain as restricted quotient bundles $\varphi^*({\cal Q})$ points of an open dense subset of the moduli space $M(X;r,d)$ of rank $r$ stable vector bundles on $X$ with degree $d$. We can extend this result to the flag varieties. For the projective spaces ${\bf P}^n$, we obtain that if $d$ is large with respect to $g$, $d\ge ng+1$, then degree $d$ embeddings $\varphi:X\to{\bf P}^n$ cover a dense open subset of the moduli space $M(X;n,(n+1)d)$ by means of the restricted tangent bundles $\varphi^*(T_{{\bf P}^n})$. This fact does not hold for restricted tangent bundles of a Grassmannian $G(r,\nu)$ with $2\le r\le\nu-2$. However, for a large degree $d$, we are able to characterize the restricted tangent bundles $\varphi^*(T_{G(r,\nu)})$ of a Grassmannian, obtaining that in general they are stable. For an elliptic curve $Y$, we show that in characteristic 0 there is a degree $d$ embedding of $Y$ in a Grassmannian with a stable restricted tangent bundle if and only if there is not a numerical restriction to its existence.