GKZ discriminant and Multiplicities
Abstract
Let $T=(\mathbb{C}^*)^k$ act on $V=\mathbb{C}^N$ faithfully and preserving the volume form, i.e. $(\mathbb{C}^*)^k \hookrightarrow \text{SL}(V)$. On the Bside, we have toric stacks $Z_W$ labelled by walls $W$ in the GKZ fan, and toric stacks $Z_{/F}$ labelled by faces of a polytope corresponding to minimal semiorthogonal decomposition (SOD) components. The Bside multiplicity $n^B_{W,F}$ is the number of times ${Coh}(Z_{/F})$ appears in the SOD of $Coh(Z_W)$. On the Aside, we have the GKZ discriminant loci components $\nabla_F \subset (\mathbb{C}^*)^k$, and its tropicalization $\nabla^{trop}_{F} \subset \mathbb{R}^k$. The Aside multiplicity $n^A_{W, F}$ is defined as the multiplicity of the tropical complex $\nabla^{trop}_{F}$ on the wall $W$. We prove that $n^A_{W,F} = n^B_{W,F}$, confirming a conjecture in KiteSegal \cite{kitesegal} inspired by \cite{aspinwall2017mirror}. Our proof is based on a lemma about the Bside SOD multiplicities, which allows us to reduce to lower dimensions just as in the Aside \cite{GKZbook}[Ch 11].
 Publication:

arXiv eprints
 Pub Date:
 June 2022
 DOI:
 10.48550/arXiv.2206.14778
 arXiv:
 arXiv:2206.14778
 Bibcode:
 2022arXiv220614778H
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematical Physics;
 14J33;
 14F08
 EPrint:
 20 pages