ACC for $F$-signature: a likely counterexample

Abstract

Let $k=\overline{{\mathbb{F}}_2}$ and let $0\ne \alpha \in k$. We present a conjecture supported by computer experimentation involving the Brenner-Monsky quartic $g_{\alpha }=\alpha x^2{y}^2+z^4+xy{z}^2+(x^3+y^3)z\in k[[x,y,z]]$. If true, this conjecture provides a formula for the Hilbert-Kunz multiplicity and $F$-signature of the family of four-dimensional hypersurfaces defined by $uv+g_{\alpha }\in k[[x,y,z,u,v]]$ which depends on $[{\mathbb{F}}_2(\alpha ):{\mathbb{F}}_2]$, giving an infinite increasing chain of strict inequalities of $F$-signatures. Additionally, we obtain for any $t\in \mathbb{N}$ a formula for the Hilbert-Kunz multiplicity and $F$-signature of the $t$-parameter family of $3t+1$-dimensional hypersurfaces defined by $uv+\sum _{i=1}^t{g}_{{\alpha }_i}(x_i,y_i,z_i)$.

Written during the 2023 SMALL REU at Williams College. 16 pages. Comments welcome