Formulas for the F-pure Threshold of Some Algebras with Straightening Laws

In positive characteristic commutative algebra there exists a numerical invariant which measures the singularities of a variety, called the $F$-pure threshold, which is the analog of another invariant called the log canonical threshold in characteristic zero. We compute the $F$-pure threshold $\fpt(\mathfrak{m})$ of the irrelevant maximal ideal $\frak{m}$ of some graded rings. In particular, we focus on graded rings that have the structure of an algebra with straightening laws, or ASL for short, which are algebras whose structure is determined by an underlying poset. The ASLs that we consider in this work include the homogeneous coordinate rings of flag varieties and the homogeneous coordinate rings of Schubert varieties inside a fixed Grassmannian. Our calculation of the $F$-pure threshold for these rings is made possible through the combinatorial properties of their ASL structure. We also compute the $F$-pure threshold of some non-ASLs such as the symmetric determinantal rings and special classes of flag varieties through other means. We also note that every instance of our computation of $\fpt(\mathfrak{m})$ relied on the computation of another invariant of graded rings, the $a$-invariant.