Finiteness of the polyhedral ℚ-codegree spectrum

Abstract

In this note we prove Fujita’s spectrum conjecture for polarized varieties in the case of Q \mathbb {Q} -Gorenstein projective toric varieties of index r 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 Q \mathbb {Q} -codegree of the polytope. Now the main theorem of the paper shows that the spectrum of the Q \mathbb {Q} -codegree is finite above any positive threshold in the class of lattice polytopes with α \alpha -canonical normal fan for any fixed α > 0 \alpha >0 .