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DEGENERACY OF TRIALITY-SYMMETRIC MORPHISMS DAVE ANDERSON
 

Summary: DEGENERACY OF TRIALITY-SYMMETRIC MORPHISMS
DAVE ANDERSON
Abstract. We define a new symmetry for morphisms of vector bundles,
called triality symmetry, and compute Chern class formulas for the degen-
eracy loci of such morphisms. In an appendix, we show how to canonically
associate an octonion algebra bundle to any rank 2 vector bundle.
1. Introduction
Let : E F be a morphism of vector bundles on a smooth variety X, of
respective ranks m and n. The rth degeneracy locus of is the set of points of
X defined by
Dr() = {x X | rk (x) r},
where (x) : E(x) F(x) is the corresponding linear map in the fibers over
x X. Such loci are ubiquitous in algebraic geometry: many interesting
varieties, from Veronese embeddings of projective spaces to Brill­Noether loci
parametrizing special divisors in Jacobians, can be realized as degeneracy loci
for appropriate maps of vector bundles. General geometric information about
degeneracy loci is therefore often useful. In particular, one can ask for Chern
class formulas for the cohomology class of Dr() in HX -- what is [Dr()]
as a polynomial in the Chern classes of E and F?
When is sufficiently general, so Dr() has expected codimension equal to

  

Source: Anderson, Dave - Department of Mathematics, University of Washington at Seattle

 

Collections: Mathematics