Fano schemes of complete intersections in toric varieties

We study Fano schemes $$\mathrm{F}_k(X)$$ for complete intersections X in a projective toric variety $$Y\subset \mathbb {P}^n$$ . Our strategy is to decompose $$\mathrm{F}_k(X)$$ into closed subschemes based on the irreducible decomposition of $$\mathrm{F}_k(Y)$$ 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 $$\mathrm{F}_k(X)$$ is zero.