Primitive collections and toric varieties

Abstract

This paper studies Batyrev's notion of primitive collection. We use primitive collections to characterize the nef cone of a quasi-projective toric variety whose fan has convex support, a result stated without proof by Batyrev in the smooth projective case. When the fan is non-simplicial, we modify the definition of primitive collection and explain how our definition relates to primitive collections of simplicial subdivisons. The paper ends with an open problem.2000 Mathematics Subject Classification. Primary 14M25

ℙ 1 $$ {\mathrm{\mathbb{P}}}^1 $$ -bundle bases and the prevalence of non-Higgsable structure in 4D F-theory models

“…It is convenient to keep all of these classes separate, however, as they act as an effective (though redundant) set of generators for the Mori cone (see e.g. [40]). To check the weak Fano condition, therefore, it suffixes to check that −K · C ≥ 0 for each of the curve classes just mentioned.…”

A Batyrev type classification of ℚ-factorial projective toric varieties

“…In particular we get the following Proposition 2.3 (Lemma 1.4 in [5], Prop. 1.9 in [9]). If P = {V P } is a primitive collection, for some P ∈ PC(Σ), then its numerical class n P is positive against every nef divisor of X(Σ), which is γ Σ ⊆ H + P .…”

“…Let us start by proving the only if condition: in fact after Thm. 1.4 in [9] one knows that every facet of the chamber γ generates a primitive collection P: e.g. by thinking P * as all the columns q of Q = G(V ) such that n • q > 0, where n is the inward primitive normal vector n to the considered facet.…”

“…Bypassing counterexample 2.31 means characterizing those PWS admitting a nef primitive collection. This is done by stressing remarks of C. Casagrande [5] and of D. Cox and C. von Renesse [9], revising the original Batyrev definition of primitive relation: § 2.1 is largely devoted to this purpose. The idea is that of dually thinking of the numerical class of a primitive relation as a hyperplane in Cl(X) ⊗ R, which we called the supporting hyperplane of the primitive collection (see Definition 2.1).…”

Moment Maps, Strict Linear Precision, and Maximum Likelihood Degree One

“…This is not surprising, since the toric variety corresponding to P is not simplicial by Proposition 8.6. Most results about primitive collections apply to simplicial toric varieties [4, §6.4], though some results hold more generally [6].…”