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Audiovisual Mathematics Library

Audiovisual Mathematics Library

Documents 14M12 2 results

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Sums of three squares and Noether-Lefschetz loci - Benoist, Olivier (Author of the conference) | CIRM H

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It is a theorem of Hilbert that a real polynomial in two variables that is nonnegative is a sum of 4 squares of rational functions. Cassels, Ellison and Pfister have shown the existence of such polynomials that are not sums of 3 squares of rational functions. In this talk, we will prove that those polynomials that may be written as sums of 3 squares are dense in the set of nonnegative polynomials. The proof is Hodge-theoretic.

11E25 ; 14Pxx ; 14D07 ; 14M12

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Orbital degeneracy loci - Benedetti, Vladimiro (Author of the conference) | CIRM H

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I will present a joint work with Sara Angela Filippini, Laurent Manivel and Fabio Tanturri (arXiv: 1704.01436). We introduce a new class of varieties, called orbital degeneracy loci. The idea is to use any orbit closure in a representation of an algebraic group to generalise the classical construction of degeneracy loci of morphisms between vector bundles, and of zero loci as well. After giving the definition of an orbital degeneracy locus, I will explain how to control the canonical bundle of these varieties: under some Gorenstein condition on the orbit closure, it is possible to construct examples of varieties with trivial canonical bundle or of Fano type. Finally, if time will permit, I will give some explicit examples of such degeneracy loci, which allow to construct many Calabi-Yau varieties of dimension three and four, and some new Fano fourfolds.[-]
I will present a joint work with Sara Angela Filippini, Laurent Manivel and Fabio Tanturri (arXiv: 1704.01436). We introduce a new class of varieties, called orbital degeneracy loci. The idea is to use any orbit closure in a representation of an algebraic group to generalise the classical construction of degeneracy loci of morphisms between vector bundles, and of zero loci as well. After giving the definition of an orbital degeneracy locus, I ...[+]

14M12 ; 14C05 ; 14M15 ; 14J60 ; 14J32 ; 14J45

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