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

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

2004-03-18

arxiv.org/abs/math/0403300v2

Mathematics

Algebraic Geometry

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

Scientific paper

In the present paper the small Quantum Cohomology ring of some Fano threefolds which are obtained as one- or two-curve blow-ups from $P^3$ or the quadric $Q^3$ 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 $b_3(X)=0$ admits a complete exceptional set of the appropriate length.

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