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Manin’s conjecture for a singular sextic del Pezzo surface

Daniel Loughran (2010)

Journal de Théorie des Nombres de Bordeaux

We prove Manin’s conjecture for a del Pezzo surface of degree six which has one singularity of type A 2 . Moreover, we achieve a meromorphic continuation and explicit expression of the associated height zeta function.

Matrix factorizations and singularity categories for stacks

Alexander Polishchuk, Arkady Vaintrob (2011)

Annales de l’institut Fourier

We study matrix factorizations of a potential W which is a section of a line bundle on an algebraic stack. We relate the corresponding derived category (the category of D-branes of type B in the Landau-Ginzburg model with potential W) with the singularity category of the zero locus of W generalizing a theorem of Orlov. We use this result to construct push-forward functors for matrix factorizations with relatively proper support.

Matrix factorizations for domestic triangle singularities

Dawid Edmund Kędzierski, Helmut Lenzing, Hagen Meltzer (2015)

Colloquium Mathematicae

Working over an algebraically closed field k of any characteristic, we determine the matrix factorizations for the-suitably graded-triangle singularities f = x a + y b + z c of domestic type, that is, we assume that (a,b,c) are integers at least two satisfying 1/a + 1/b + 1/c > 1. Using work by Kussin-Lenzing-Meltzer, this is achieved by determining projective covers in the Frobenius category of vector bundles on the weighted projective line of weight type (a,b,c). Equivalently, in a representation-theoretic context,...

Maximal rationally connected fibrations and movable curves

Luis E. Solá Conde, Matei Toma (2009)

Annales de l’institut Fourier

A well known result of Miyaoka asserts that a complex projective manifold is uniruled if its cotangent bundle restricted to a general complete intersection curve is not nef. Using the Harder-Narasimhan filtration of the tangent bundle, it can moreover be shown that the choice of such a curve gives rise to a rationally connected foliation of the manifold. In this note we show that, conversely, a movable curve can be found so that the maximal rationally connected fibration of the manifold may be recovered...

Minimal resolution of general stable rank-2 vector bundles on P 2

Carla Dionisi, Marco Maggesi (2003)

Bollettino dell'Unione Matematica Italiana

We study general elements of moduli spaces M P 2 ⁢ 2 , c 1 , c 2 of rank-2 stable holomorphic vector bundles on P 2 and their minimal free resolutions. Incidentally, a quite easy proof of the irreducibility of M P 2 ⁢ 2 , c 1 , c 2 is shown.

Minimal sections of conic bundles

Atanas Iliev (1999)

Bollettino dell'Unione Matematica Italiana

Sia p : X → P 2 un fibrato in coniche standard con curva discriminante Δ di grado d . La varietà delle sezioni minime delle superfici p - 1 ⁢ C , dove C è una curva di grado d - 3 , si spezza in due componenti C + e C - . Si prova che, mediante la mappa di Abel-Jacobi Φ , una di queste componenti domina la Jacobiana intermedia J ⁢ X , mentre l'altra domina il divisore theta Θ ⊂ J ⁢ X . Questi risultati vengono applicati ad alcuni threefold di Fano birazionalmente equivalenti a un fibrato in coniche. In particolare si prova che il generico...

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