Computing the quantum cohomology of some Fano threefolds and its semisimplicity
Bollettino della Unione matematica italiana, Série 8, 7B (2004) no. 2, pp. 511-517
Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica
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.
@article{BUMI_2004_8_7B_2_a15,
author = {Ciolli, Gianni},
title = {Computing the quantum cohomology of some {Fano} threefolds and its semisimplicity},
journal = {Bollettino della Unione matematica italiana},
pages = {511--517},
publisher = {mathdoc},
volume = {Ser. 8, 7B},
number = {2},
year = {2004},
zbl = {1182.14058},
mrnumber = {MR2072951},
language = {en},
url = {http://geodesic.mathdoc.fr/item/BUMI_2004_8_7B_2_a15/}
}
TY - JOUR AU - Ciolli, Gianni TI - Computing the quantum cohomology of some Fano threefolds and its semisimplicity JO - Bollettino della Unione matematica italiana PY - 2004 SP - 511 EP - 517 VL - 7B IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/BUMI_2004_8_7B_2_a15/ LA - en ID - BUMI_2004_8_7B_2_a15 ER -
Ciolli, Gianni. Computing the quantum cohomology of some Fano threefolds and its semisimplicity. Bollettino della Unione matematica italiana, Série 8, 7B (2004) no. 2, pp. 511-517. http://geodesic.mathdoc.fr/item/BUMI_2004_8_7B_2_a15/