The structure of the cohomology ring of the filt schemes

2003/01/17 by Takuro Mochizuki, Mochizuki, Takuro
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #Polynomial and algebraic computation #math.AG #math.AT #msc:14F45 #msc:57R17

paper · pdf · doi:10.48550/arxiv.math/0301184

arxiv created 2003/01/17 · arxiv updated 2009/11/30

Abstract

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^{⊕ r} with length n. We obtain a filtration on H^∗(\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^∗(\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.

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