Toric residue and combinatorial degree

Soprounov, Ivan

doi:10.1090/S0002-9947-04-03770-5

Toric residue and combinatorial degree
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Trans. Amer. Math. Soc. 357 (2005), 1963-1975 Request permission

Abstract:

Consider an $n$-dimensional projective toric variety $X$ defined by a convex lattice polytope $P$. David Cox introduced the toric residue map given by a collection of $n+1$ divisors $(Z_0,\dots ,Z_n)$ on $X$. In the case when the $Z_i$ are $\mathbb {T}$-invariant divisors whose sum is $X\setminus \mathbb {T}$, the toric residue map is the multiplication by an integer number. We show that this number is the degree of a certain map from the boundary of the polytope $P$ to the boundary of a simplex. This degree can be computed combinatorially. We also study radical monomial ideals $I$ of the homogeneous coordinate ring of $X$. We give a necessary and sufficient condition for a homogeneous polynomial of semiample degree to belong to $I$ in terms of geometry of toric varieties and combinatorics of fans. Both results have applications to the problem of constructing an element of residue one for semiample degrees.

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