Computing the quantum cohomology of some Fano threefolds and its semisimplicity

Gianni Ciolli

Bollettino dell'Unione Matematica Italiana (2004)

  • Volume: 7-B, Issue: 2, page 511-517
  • ISSN: 0392-4041

Abstract

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We compute explicit presentations for the small Quantum Cohomology ring of some Fano threefolds which are obtained as one- or two-curve blow-ups from or the smooth quadric. Systematic usage of the associativity property of quantum product implies that 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 with admits a complete exceptional set of the appropriate length. Details are contained in the preprint [4] and will be published elsewhere.

How to cite

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Ciolli, Gianni. "Computing the quantum cohomology of some Fano threefolds and its semisimplicity." Bollettino dell'Unione Matematica Italiana 7-B.2 (2004): 511-517. <http://eudml.org/doc/195165>.

@article{Ciolli2004,
abstract = {We compute explicit presentations for the small Quantum Cohomology ring of some Fano threefolds which are obtained as one- or two-curve blow-ups from $\mathbb\{P\}^\{3\}$ or the smooth quadric. Systematic usage of the associativity property of quantum product implies that 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. Details are contained in the preprint [4] and will be published elsewhere.},
author = {Ciolli, Gianni},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {511-517},
publisher = {Unione Matematica Italiana},
title = {Computing the quantum cohomology of some Fano threefolds and its semisimplicity},
url = {http://eudml.org/doc/195165},
volume = {7-B},
year = {2004},
}

TY - JOUR
AU - Ciolli, Gianni
TI - Computing the quantum cohomology of some Fano threefolds and its semisimplicity
JO - Bollettino dell'Unione Matematica Italiana
DA - 2004/6//
PB - Unione Matematica Italiana
VL - 7-B
IS - 2
SP - 511
EP - 517
AB - We compute explicit presentations for the small Quantum Cohomology ring of some Fano threefolds which are obtained as one- or two-curve blow-ups from $\mathbb{P}^{3}$ or the smooth quadric. Systematic usage of the associativity property of quantum product implies that 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. Details are contained in the preprint [4] and will be published elsewhere.
LA - eng
UR - http://eudml.org/doc/195165
ER -

References

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