Canonical Lattices of Integer Relations Associated to Rational Fans: Wall Generation and a Two-Step Support Filtration

Abstract

We study the lattice $L_{\mathrm{rel}}(\Sigma )=\ker ({\mathbb{Z}}^{\Sigma (1)}\to N)$ of integer relations among the primitive ray generators of a rational fan $\Sigma$, from an intrinsic, coordinate-free point of view. For each cone $\tau \in \Sigma$ we introduce the \emph{star-supported} sublattice $L_{\mathrm{rel}}(\mathrm{Star}(\tau ))$ of relations whose support lies in the star of $\tau$, and we organize these by codimension into a support filtration $F_{\bullet }{L}_{\mathrm{rel}}(\Sigma )$. Our main result is a sharp local generation theorem: for a complete fan the relation lattice is generated \emph{integrally} by the relations supported on the stars of walls (codimension-one cones). Equivalently, the support filtration collapses after a single step, $F_1{L}_{\mathrm{rel}}(\Sigma )=L_{\mathrm{rel}}(\Sigma )$. This is an intrinsic repackaging of the classical wall (wall-crossing) relations that generate the group of numerically trivial classes on a complete toric variety. We make the resulting two-step structure precise: for simplicial fans one has $0=F_0⊊F_1=L_{\mathrm{rel}}(\Sigma )$, while for general fans $F_0$ records the intrinsic relations of non-simplicial maximal cones and $F_1$ adds exactly the wall relations. We prove functoriality of $L_{\mathrm{rays}}$ and $L_{\mathrm{rel}}$ under fan isomorphisms and ray-preserving subdivisions, deduce that every primitive collection of size $m$ is wall-generated, and illustrate the theory on ${\mathbb{P}}^2\times {\mathbb{P}}^1$, products of projective lines, weighted projective spaces, and the (non-simplicial) fan over a cube. We are careful throughout to distinguish what the filtration does and does not detect, correcting a natural but false expectation that support-codimension yields a strictly increasing multi-step invariant.

8 pages