The structure of the cohomology ring of the filt schemes
Abstract
Description
Let $C$ be a smooth projective curve over the complex number field $\cnum$. We investigate the structure of the cohomology ring of the quot schemes $\Quot(r,n)$, i.e., the moduli scheme of the quotient sheaves of $\nbigo_C^{\oplus r}$ with length $n$. We obtain a filtration on $H^{\ast}(\Quot(r,n))$, whose associated graded ring has a quite simple structure. As a corollary, we obtain a small generator of the ring. We also obtain a precise combinatorial description of $H^{\ast}(\Quot(r,n))$ itself.
For that purpose, we consider the complete filt schemes and we use a `splitting principle'. A complete filt scheme has an easy geometric description: a sequence of bundles of projective spaces. Thus the cohomology ring of the complete filt schemes are easily determined. In the cohomology ring of complete filt schemes, we can do some calculations for the cohomology ring of quot schemes.
Keywords: Quot scheme, cohomology ring, torus action, splitting principle, symmetric function.