Severi varieties and their varieties of reductions (original) (raw)
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Severi varieties over arbitrary fields
Our main aim is to provide a uniform geometric characterization of the Severi varieties over arbitrary fields, i.e. the quadric Veronese varieties in 5-dimensional projective spaces, the Segre varieties in 8-dimensional projective spaces, the line Grassmannians in 14-dimensional projective spaces, and the 26-dimensional exceptional varieties of type E 6. Our theorem can be regarded as a counterpart over arbitrary fields of the classification of smooth complex algebraic Severi varieties. Our axioms are based on an old characterization of finite quadric Veronese varieties by Mazzocca and Melone, and our results can be seen as a far-reaching generalization of Mazzocca and Melone's approach that characterizes finite varieties by requiring just the essential algebraic-geometric properties. We allow just enough generalization to capture the Severi varieties and some related varieties, over an arbitrary field. The proofs just use projective geometry.
Geometry of the Severi variety
AMERICAN MATHEMATICAL SOCIETY, 1988
ABSTRACT. This paper is concerned with the geometry of the Severi variety W parametrizing plane curves of given degree and genus, and specifically with the relations among various divisor classes on W. Two types of divisor classes on W are described: those that come from the ...
On Severi varieties as intersections of a minimum number of quadrics
Cubo (Temuco)
Let \({\mathscr{V}}\) be a variety related to the second row of the Freudenthal-Tits Magic square in \(N\)-dimensional projective space over an arbitrary field. We show that there exist \(M\leq N\) quadrics intersecting precisely in \({\mathscr{V}}\) if and only if there exists a subspace of projective dimension \(N-M\) in the secant variety disjoint from the Severi variety. We present some examples of such subspaces of relatively large dimension. In particular, over the real numbers we show that the Cartan variety (related to the exceptional group \({E_6}\)\((\mathbb R)\)) is the set-theoretic intersection of 15 quadrics.
Journal de l’École polytechnique — Mathématiques, 2021
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Fano Symmetric Varieties with Low Rank
Publications of the Research Institute for Mathematical Sciences, 2012
The symmetric projective varieties of rank one are all smooth and Fano by a classic result of Akhiezer. We classify the locally factorial (respectively smooth) projective symmetric G-varieties of rank 2 which are Fano. When G is semisimple we classify also the locally factorial (respectively smooth) projective symmetric G-varieties of rank 2 which are only quasi-Fano. Moreover, we classify the Fano symmetric G-varieties of rank 3 obtainable from a wonderful variety by a sequence of blow-ups along G-stable varieties. Finally, we classify the Fano symmetric varieties of arbitrary rank which are obtainable from a wonderful variety by a sequence of blow-ups along closed orbits.
Tangential varieties of Segre varieties
2011
We determine the minimal generators of the ideal of the tangential variety of a Segre-Veronese variety, as well as the decomposition into irreducible GL-representations of its homogeneous coordinate ring. In the special case of a Segre variety, our results confirm a conjecture of Landsberg and Weyman.
Deformations of canonical pairs and Fano varieties
Journal für die reine und angewandte Mathematik (Crelles Journal), 2000
This paper is devoted to the study of various aspects of deformations of log pairs, especially in connection to questions related to the invariance of singularities and log plurigenera. In particular, using recent results from the minimal model program, we obtain an extension theorem for adjoint divisors in the spirit of Siu and Kawamata and more recent works of Hacon and M c Kernan. Our main motivation however comes from the study of deformations of Fano varieties. Our first application regards the behavior of Mori chamber decompositions in families of Fano varieties: we prove that, in the case of mild singularities, such decomposition is rigid under deformation when the dimension is small. We then turn to analyze deformation properties of toric Fano varieties, and prove that every simplicial toric Fano variety with at most terminal singularities is rigid under deformations (and in particular is not smoothable, if singular).