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Computing the quantum cohomology of some Fano threefolds and its semisimplicity

Gianni Ciolli (2004)

Bollettino dell'Unione Matematica Italiana

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 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...

Construction d’hypersurfaces irréductibles avec lieu singulier donné dans ℂ n

Jean-Pierre Demailly (1980)

Annales de l'institut Fourier

Étant donné un ensemble analytique S de codimension ≥ 2 dans C n , nous construisons des hypersurfaces irréductibles de lieu singulier S , avec contrôle de la croissance. À partir d’un contre-exemple au problème de Bezout transcendant, dû à M. Cornalba et B. Shiffman, nous montrons l’existence d’une courbe irréductible d’ordre 0 dans C 2 , dont le lieu singulier est d’ordre infini. Nous étudions également en application certaines propriétés arithmétiques de l’anneau de convolution ℰ ' ( R n ) .

Contractions of smooth varieties. II. Computations and applications

Marco Andreatta, Jarosław A. Wiśniewski (1998)

Bollettino dell'Unione Matematica Italiana

Una contrazione su una varietà proiettiva liscia X è data da una mappa φ : X → Z propria, suriettiva e a fibre connesse in una varietà irriducibile normale Z . La contrazione si dice di Fano-Mori se inoltre - K X è φ -ampio. Nel lavoro, naturale seguito e completamento delle ricerche introdotte in [A-W3], si studiano diversi aspetti delle contrazioni di Fano-Mori attraverso esempi (capitolo 1) e teoremi di struttura (capitoli 3 e 4). Si discutono anche alcune applicazioni allo studio di morfismi birazionali propri...

Convex bodies associated to linear series

Robert Lazarsfeld, Mircea Mustață (2009)

Annales scientifiques de l'École Normale Supérieure

In his work on log-concavity of multiplicities, Okounkov showed in passing that one could associate a convex body to a linear series on a projective variety, and then use convex geometry to study such linear systems. Although Okounkov was essentially working in the classical setting of ample line bundles, it turns out that the construction goes through for an arbitrary big divisor. Moreover, this viewpoint renders transparent many basic facts about asymptotic invariants of linear series, and opens...

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