The absolutely Koszul property of Veronese subrings and Segre products

Absolutely Koszul algebras are a class of rings over which any finite graded module has a rational Poincaré series. We provide a criterion to detect non-absolutely Koszul rings. Combining the criterion with Macaulay2 computations, we identify large families of Veronese subrings and Segre products of polynomial rings which are not absolutely Koszul. In particular, we classify completely the absolutely Koszul algebras among Segre products of polynomial rings, when the base field has characteristic 0.     

F-Thresholds versus a-Invariants for Standard Graded Toric Rings

In this paper, we establish inequalities between F-pure thresholds or F-thresholds and classical a-invariants. Moreover, we characterize Gorenstein toric rings in terms of F-pure thresholds and a-invariants.     

Beniamino Segre

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Beniamino Segre

     

Rigidity and holomorphic Segre transversality for holomorphic Segre maps

Let and denote the complexifications of Heisenberg hypersurfaces in and , respectively. We show that non-degenerate holomorphic Segre mappings from into with possess a partial rigidity property. As an application, we prove that the holomorphic Segre non-transversality for a holomorphic Segre map from into with propagates along Segre varieties. We also give an example showing that this propagation property of holomorphic Segre transversality fails when N > 2n − 2.     

Toric cubes

A toric cube is a subset of the standard cube defined by binomial inequalities. These basic semialgebraic sets are precisely the images of standard cubes under monomial maps. We study toric cubes from the perspective of topological combinatorics. Explicit decompositions as CW-complexes are constructed. Their open cells are interiors of toric cubes and their boundaries are subcomplexes. The motivating example of a toric cube is the edge-product space in phylogenetics, and our work generalizes results known for that space.     

Syzygy algebras for Segre embeddings

We describe the syzygy spaces for the Segre embedding ?(U) × ?(V) ? ?(U ? V) in terms of representations of GL(U) × GL(V) and construct the minimal resolutions of the sheaves (a, b) in D(?(U ? V)) for a ? ?dim(U) and b ? ?dim(V). We also prove a property of multiplication in syzygy spaces of the Segre embedding.     

L'opera geometrica di Corrado Segre

Nato a Saluzzo ai 20 di agosto del 1863; laureato in matematica dalla Università di Torino il 1° luglio 1883. Durante l'anno 1883–84 fu in questo Ateneo assistente alla cattedra di algebra complementare e geometria analitica (prof.E. D'Ovidio) e nel seguente prestò il prescritto servizio militare; nel triennio 1885–88 occupò il posto di assistente alla cattedra di geometria proiettiva e descrittiva (prof.G. Bruno) facendo le lezioni della prima di tali materie. MortoF. Faà di Bruno, ilD'Ovidio passò dall'incarico della Geometria superiore a quello dell'Analisi superiore ed ilSegre fu nominato (15 novembre 1888) per concorso professore straordinario di quella disciplina; quattro anni appresso venne promosso ordinario. Durante tre trienni, cioè dal 1909 al 1918 fu preside della Facoltà di Scienze della Università di Torino; nel biennio 1895–97 ebbe anche l'incarico dell'insegnamento della fisica matematica e dal 1918 al 1922 quello delle conferenze di magistero in matematica. Un morbo repentino ed inesorabile lo spense addì 18 maggio 1924.     

On syzygies of Segre embeddings

We study the syzygies of the ideals of the Segre embeddings. Let , ; we prove that the line bundle on the ( copies) satisfies Property of Green-Lazarsfeld if and only if . Besides we prove that if we have a projective variety not satisfying Property for some , then the product of it with any other projective variety does not satisfy Property . From this we also deduce other corollaries about syzygies of Segre embeddings.

     

Remarks on the Segre cubic

The Igusa quartic 3-fold is the moduli space of Abelian surfaces with the level two structure, and it is known that the tangent hyperplanes cut out Kummer surfaces. In this note, we show similar results for the Segre cubic.Received: 18 March 2002     

Toric Rigid Spaces

This paper gives a method to construct rigid spaces, which is similar to the method used to construct toric schemes.