Bi-Co Mathematics Colloquium with Dr. Takayuki Hibi

Title: "Pick’s Formula and Castelnuovo Polytopes" Abstract: Let P ⊂ Rd be a lattice polytope of dimension d. Let b(P) denote the number of lattice points belonging to the boundary of P and c(P) that to the interior of P. It follows from the lower bound theorem of Ehrhart polynomials that, when c(P) > 0, one has (1) vol(P) ≥ ((d − 1) · b(P) + d · c(P) − d2 + 2)/d!, where vol(P) is the (Lebesgue) volume of P. Pick’s formula guarantees that, when d = 2, the inequality (1) is an equality. One calls P Castelnuovo if c(P) > 0 and if the equal sign holds in (1). In the talk, after a quick introduction to Ehrhart theory of lattice polytopes, to explain the reason why one calls Castelnuovo, a historical background on polarized toric varieties will be briefly presented. No special knowledge will be required to understand the talk.