ACC for F-signature: a likely counterexample

2023/09/14 by Adams, Clay, Sandstrom, Theodore J., Simpson, Austyn
#Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2309.07901

Abstract

Let \mathscrk=\mathbbF₂ and let 0≠α∈ \mathscrk. We present a conjecture supported by computer experimentation involving the Brenner-Monsky quartic g_α=αx²y²+z⁴+xyz²+(x³+y³)z∈ \mathscrk[[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_α∈ \mathscrk[[x,y,z,u,v]] which depends on [\mathbbF₂(α):\mathbbF₂], giving an infinite increasing chain of strict inequalities of F-signatures. Additionally, we obtain for any t∈ℕ a formula for the Hilbert-Kunz multiplicity and F-signature of the t-parameter family of 3t+1-dimensional hypersurfaces defined by uv+∑_i=1^t g_αi(x_i,y_i,z_i).

Related

• Hilbert-Kunz multiplicities and the F-signature
• F-signature functions of diagonal hypersurfaces
• Limits of F-invariants and Riemann-Stieltjes integral
• F-signature of pairs: Continuity, p-fractals and minimal log discrepancies
• Hilbert-Kunz multiplicity and F-signature can disagree