Vector bundles on proper toric 3-folds and certain other schemes

2014/07/21 by Markus Perling, Perling, Markus, Stefan Schroeer +1
Mathematics · #14J60 #14M25 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14J60 #msc:14M25

paper · pdf · doi:10.48550/arxiv.1407.5443

28 pages, minor changes, to appear in Trans. Amer. Math. Soc

arxiv created 2015/08/24 · arxiv updated 2015/08/25

Abstract

We show that a proper algebraic n-dimensional scheme Y admits nontrivial vector bundles of rank n, even if Y is non-projective, provided that there is a modification containing a projective Cartier divisor that intersects the exceptional locus in only finitely many points. Moreover, there are such vector bundles with arbitrarily large top Chern number. Applying this to toric varieties, we infer that every proper toric threefold admits such vector bundles of rank three. Furthermore, we describe a class of higher-dimensional toric varieties for which the result applies, in terms of convexity properties around rays.

Related

• Vector bundles on toric varieties
• Exceptional Collections of Line Bundles for Smooth Toric Fano Surfaces and Threefolds
• Smooth projective toric varieties whose nontrivial nef line bundles are big
• Derived category of toric varieties with Picard number three
• A note on stable toric sheaves of low rank