Finiteness of the polyhedral $\mathbb {Q}$-codegree spectrum

Abstract:

In this note we prove Fujita’s spectrum conjecture for polarized varieties in the case of $\mathbb {Q}$-Gorenstein projective toric varieties of index $r$. The theorem follows from a combinatorial result using the connection between lattice polytopes and polarized projective toric varieties. By this correspondence the spectral value of the polarized toric variety equals the $\mathbb {Q}$-codegree of the polytope. Now the main theorem of the paper shows that the spectrum of the $\mathbb {Q}$-codegree is finite above any positive threshold in the class of lattice polytopes with $\alpha$-canonical normal fan for any fixed $\alpha >0$.