On the Quantum Cohomology of some Fano threefolds and a conjecture of Dubrovin

2004/03/18 by Gianni Ciolli, Ciolli, Gianni · 1 citation
Mathematics · #14J45 (Secondary) #14N35 (Primary) #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14J45 #msc:14N35

paper · pdf · doi:10.48550/arxiv.math/0403300

15 pages, 3 tables. In v2 two small mistakes were corrected: a missing hypothesis in the citation of Theorem 6 and a wrong bibliographic citation

arxiv created 2004/03/23 · arxiv updated 2009/12/01

Abstract

In the present paper the small Quantum Cohomology ring of some Fano threefolds which are obtained as one- or two-curve blow-ups from P³ or the quadric Q³ is explicitely computed. Because of systematic usage of the associativity property of quantum product only a very small and enumerative subset of Gromov-Witten invariants is needed. Then, for these threefolds the Dubrovin conjecture on the semisimplicity of Quantum Cohomology is proven by checking the computed Quantum Cohomology rings and by showing that a smooth Fano threefold X with b₃(X)=0 admits a complete exceptional set of the appropriate length.

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