F-singularities of polynomials with square-free support

TL;DR

The paper addresses whether hypersurfaces defined by irreducible square-free supported polynomials have F-rational singularities in characteristic $p>0$, and extends this to non-irreducible complete intersections while removing the algebraically closed assumption. It develops a framework combining geometric irreducibility, strong F-regularity, and reduction techniques (Fedder–Glassbrenner criteria) to show that $S/(f_1, dots,f_t)$ is geometrically F-rational whenever $f$ is square-free supported with irreducible factors $f_i$. It also provides a precise link between F-singularity invariants and Hilbert–Samuel multiplicities by computing the defect of the F-pure threshold: $ ext{dfpt}(R)= ext{e}(R)-t$ for $R=S/(f_1, dots,f_t)$ and $ ext{dfpt}(R)= ext{e}(R)-1$ when $f$ is irreducible. These results yield characteristic-zero rational singularities for the corresponding hypersurfaces and extend BMW’s results beyond algebraic closures, with applications to related polynomial families and Feynman diagram polynomials.

Abstract

We show that the intersection of the irreducible components of a hypersurface defined by a polynomial with square-free support has F-rational singularities in characteristic $p>0$. As a consequence, we obtain that hypersurfaces defined by irreducible polynomials with square-free support have F-rational singularities, positively answering a question of Bath, Mustaţă, and Walther.

Theorems & Definitions (35)

• Theorem B: Theorem \ref{['ThmReducible']}
• Theorem C: Corollary \ref{['CorSqfreeModification']}
• Theorem D: Theorem \ref{['ThmDfptOne']} & Corollary \ref{['CorDfptMany']}
• Definition 2.2
• Remark 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Lemma 2.6