Moduli of Abelian Varieties, Vinberg θ-Groups, and Free Resolutions

2012/03/31 by Laurent Gruson, Steven V Sam, Jerzy Weyman · 3 citations
Mathematics · #Abelian group #Advanced Algebra and Geometry #Algebra over a field #Algebraic Geometry and Number Theory #Biology #Combinatorics #Degeneracy (biology) #Homogeneous #Homotopy and Cohomology in Algebraic Topology #Mathematics #Moduli #Moduli space #Physics #Political science #Pure mathematics #Representation (politics) #Representation theory #Vector bundle #math.AC #math.AG #math.RT #msc:13D02 #msc:14K05 #msc:14L24 #msc:15A72

paper · pdf · doi:10.1007/978-1-4614-5292-8_13

published as Commutative Algebra (edited by Irena Peeva), 419-469, Springer, 2013 · 41 pages, uses tabmac.sty, Dedicated to David Eisenbud on the occasion of his 65th birthday; v2: fixed some typos and added references

openalex publication_date 2012/12/16 · arxiv created 2013/01/21 · arxiv updated 2013/03/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We present a systematic approach to studying the geometric aspects of Vinberg theta-representations. The main idea is to use the Borel-Weil construction for representations of reductive groups as sections of homogeneous bundles on homogeneous spaces, and then to study degeneracy loci of these vector bundles. Our main technical tool is to use free resolutions as an "enhanced" version of degeneracy loci formulas. We illustrate our approach on several examples and show how they are connected to moduli spaces of Abelian varieties. To make the article accessible to both algebraists and geometers, we also include background material on free resolutions and representation theory.

Citations

• Vector bundles on curves and generalized theta functions: recent results and open problems
• The tangent space to the moduli space of vector bundles on a curve and the singular locus of the theta divisor of the jacobian
• VECTOR BUNDLES, DUALITIES AND CLASSICAL GEOMETRY ON A CURVE OF GENUS TWO
• Enriques' classification of surfaces in char.p, III
• Enriques' Classification of Surfaces in Char. p , II
• Heisenberg invariant quartics and SU_C(2) for a curve of genus four
• Commutative Algebra

Cited by

• On linear sections of the spinor tenfold II
• Invariant theory of ∧3(9) and genus-2 curves
• Vector bundles on genus 2 curves and trivectors

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• Stable reductive varieties I: Affine varieties
• The transformation laws of algebraic theta functions
• Logarithmic abelian varieties, Part VII: Moduli