Rational Curves on Moduli Spaces of Vector Bundles

2020/07/20 by Mustopa, Yusuf, Bigas, Montserrat Teixidor i
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2007.10511

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 ≥ 1 there are \rm gcd(r, °L) unobstructed components. In addition, if k is divisible by r₁(r-r₁)(g-1) for 1≤ r₁≤ r-1, there is an additional obstructed component of the expected dimension for each such r₁. 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.

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