Author:
Ilten Nathan,Kelly Tyler L.
Abstract
AbstractWe study Fano schemes $$\mathrm{F}_k(X)$$
F
k
(
X
)
for complete intersections X in a projective toric variety $$Y\subset \mathbb {P}^n$$
Y
⊂
P
n
. Our strategy is to decompose $$\mathrm{F}_k(X)$$
F
k
(
X
)
into closed subschemes based on the irreducible decomposition of $$\mathrm{F}_k(Y)$$
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 $$\mathrm{F}_k(X)$$
F
k
(
X
)
is zero.
Publisher
Springer Science and Business Media LLC
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