On lower bounds for the F-pure threshold of equigenerated ideals

Abstract

Let $k$ be a field of positive characteristic and $R=k[x_0,\dots ,x_n]$. We consider ideals $I\subseteq R$ generated by homogeneous polynomials of degree $d$. Takagi and Watanabe proved that $\mathrm{fpt}(I)\ge \mathrm{height}(I)/d$; we classify ideals $I$ for which equality is attained. Inspired by a result of de Fernex, Ein, and Mustaţă, we give a lower bound on $\mathrm{fpt}(I)$ in terms of the height of $\tau (I^{\mathrm{fpt}(I)})$.

v1: 14 pages, comments are welcome. v2: Improvements to Theorem B. Theorem C is no longer necessary and has been removed from this article, but will appear in future work. Conjecture 5.4 from v1 has been resolved in the negative; see Example 4.4