Rational Curves on Moduli Spaces of Vector Bundles
Abstract
We completely describe the components of expected dimension of the Hilbert Scheme of rational curves of fixed degree $k$ in the moduli space ${\rm SU}_{C}(r,L)$ of semistable vector bundles of rank $r$ and determinant $L$ on a curve $C$. We show that for every $k \geq 1$ there are ${\rm gcd}(r, °L)$ unobstructed components. In addition, if $k$ is divisible by $r_1(rr_1)(g1)$ for $1\le r_1\le r1$, there is an additional obstructed component of the expected dimension for each such $r_1$. We construct families of obstructed components and show that their generic point is not the generic vector bundle of given rank and determinant. Finally, we also obtain an upper bound on the degree of rational connectedness of ${\rm SU}_{C}(r,L)$ which is linear in the dimension.
 Publication:

arXiv eprints
 Pub Date:
 July 2020
 arXiv:
 arXiv:2007.10511
 Bibcode:
 2020arXiv200710511M
 Keywords:

 Mathematics  Algebraic Geometry
 EPrint:
 18 pages, comments welcome