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Let $X$ be a smooth projective surface, $K$ the canonical divisor, $H$ a very ample divisor and $M_{H} (c_{1}, c_{2})$ the moduli space of rank-two vector bundles, $H$ -stable with Chern classes $c_{1}$ and $c_{2}$ . We prove that, if there exists $c_{1}^{'}$ such that $c_{1}$ is numerically equivalent to $2 c_{1}^{'}$ and if $c_{2} - \frac{1}{4} c_{1}^{2}$ is even, greater or equal to $H^{2} + H K + 4$ , then there is no Poincaré bundle on $M_{H} (c_{1}, c_{2}) \times X$ . Conversely, if there exists $c_{1}^{'}$ such that the number $c_{1}^{'} \cdot c_{1}$ is odd or if $\frac{1}{2} c_{1}^{2} - \frac{1}{2} c_{1} \cdot K - c_{2}$ is odd, then there exists a Poincaré bundle on $M_{H} (c_{1}, c_{2}) \times X$ .

Points at Infinity on the Fermat Curves

D.E. Rohrlich (1977)

Inventiones mathematicae

Points rationnels des courbes génériques de $ℙ^{3}$ . I

N. Mestrano (1985)

Bulletin de la Société Mathématique de France

Points rationnels des courbes génériques de $ℙ^{3}$ . II

N. Mestrano (1988)

Bulletin de la Société Mathématique de France

Points rationnels et groupes fondamentaux : applications de la cohomologie $p$ -adique

Antoine Chambert-loir (2002/2003)

Séminaire Bourbaki

Je présenterai des résultats de T. Ekedahl et H. Esnault sur les variétés projectives lisses sur un corps de caractéristique strictement positive, disons $p$ , dont deux points peuvent être liés par une chaîne de courbes rationnelles, par exemple faiblement unirationnelles, ou de Fano. Notamment : 1) sur un corps fini, de telles variétés ont un point rationnel, résultat qui généralise le théorème de Chevalley-Warning ; 2) sur un corps algébriquement clos, de telles variétés ont un groupe fondamental...

Polar classes of singular varieties

Ragni Piene (1978)

Annales scientifiques de l'École Normale Supérieure

Polar quotients and singularities at infinity of polynomials in two complex variables

Arkadiusz Płoski (2002)

Annales Polonici Mathematici

Using the notion of the maximal polar quotient we characterize the critical values at infinity of polynomials in two complex variables. As an application we give a necessary and sufficient condition for a family of affine plane curves to be equisingular at infinity.

Polarized Mixed Hodge Structures and the Local Monodromy of a Variation of Hodge Structure.

Eduardo Cattani, Aroldo Kaplan (1982)

Inventiones mathematicae

Polarized surfaces $(X, L)$ with $g (L) = q (X) + m$ and $h^{0} (L) \geq m + 2$

Yoshiaki Fukuma (2003)

Rendiconti del Seminario Matematico della Università di Padova

Polynomial periodicity for Betti numbers of covering surfaces.

Eriko Hironaka (1992)

Inventiones mathematicae

Polynomial structures and generic Torelli for projective hypersurfaces

David A. Cox, Mark L. Green (1990)

Compositio Mathematica

Polytopal linear algebra.

Bruns, Winfried, Gubeladze, Joseph (2002)

Beiträge zur Algebra und Geometrie

Positive Curves in Polarized Manifolds.

L. Badescu, M.C. Beltrametti (1997)

Manuscripta mathematica

Positivity and Kleiman transversality in equivariant $K$ -theory of homogeneous spaces

Dave Anderson, Stephen Griffeth, Ezra Miller (2011)

Journal of the European Mathematical Society

We prove the conjectures of Graham–Kumar [GrKu08] and Griffeth–Ram [GrRa04] concerning the alternation of signs in the structure constants for torus-equivariant $K$ -theory of generalized ﬂag varieties $G / P$ . These results are immediate consequences of an equivariant homological Kleiman transversality principle for the Borel mixing spaces of homogeneous spaces, and their subvarieties, under a natural group action with ﬁnitely many orbits. The computation of the coefficients in the expansion of the equivariant...

Positivity of Schur function expansions of Thom polynomials

Piotr Pragacz, Andrzej Weber (2007)

Fundamenta Mathematicae

Combining the approach to Thom polynomials via classifying spaces of singularities with the Fulton-Lazarsfeld theory of cone classes and positive polynomials for ample vector bundles, we show that the coefficients of the Schur function expansions of the Thom polynomials of stable singularities are nonnegative with positive sum.

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