Fano schemes of complete intersections in toric varieties - Ilten, Nathan; Kelly, Tyler L. (Mathematische Zeitschrift, 2022)
We study Fano schemes $$F_k(X)$$for complete intersections X in a projective toric variety $$Y\subset \mathbb {P}^n$$. 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.