Section 4.2: Projective varieties and Lorentzian polynomials

We will start with ample divisors and discuss a crucial result—Kleiman’s theorem: every nef divisor is a limit of ample divisors and then give a brief proof of the main theorem. Following that, we will delve deeper into the geometric meaning of this polynomial, why we care about it, and the significance of our main theorem.

This week we review all the necessary definitions and theorems from the beginning, and complete the proof of the main theorem: If \(H_1, \dots, H_n\) are nef divisors on \(Y\), then \(\text{vol}_H(w)=(w_1H_1 + \dots + w_nH_n)^d\) is a Lorentzian polynomial.

In addition to this, we clarify Kleiman’s theorem, which equips \(\text{Div}(Y)/\text{numerical equivalence}\) with an Euclidean topology and tells us that the nef cone is the closure of the ample cone.