Fano Schemes of Complete Intersections in Toric Varieties

2019/10/12 by Ilten, Nathan, Kelly, Tyler L.
#14C05 #14M10 #14M25 #14N15 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1910.05593

Abstract

We study Fano schemes F_k(X) for complete intersections X in a projective toric variety Y⊂ ℙⁿ. Our strategy is to decompose F_k(X) into closed subschemes based on the irreducible decomposition of F_k(Y) as studied by Ilten and Zotine. We define the expected dimension for these subschemes, which always gives a lower bound on the actual dimension. Under additional assumptions, we show that these subschemes are non-empty and smooth of the expected dimension. Using tools from intersection theory, we can apply these results to count the number of linear subspaces in X when the expected dimension of F_k(X) is zero.

Related

• On Fano schemes of linear spaces of general complete intersections
• On Fano schemes of complete intersections
• Numerical invariants of Fano schemes of linear subspaces on complete intersections
• Notes on Fano varieties of complete intersections
• The anticanonical complex for non-degenerate toric complete intersections