\(F\)-pure thresholds and \(F\)-jumping exponents in dimension two
Let \(K\) be a perfect field of characteristic \(p>0\), \(X\) a smooth \(K\)--variety and \(D\) an effective divisor on \(X\). The \(F\)--pure threshold of \((X, D)\) at \(x\in X\) is defined by \(\text{fpt}_x(X,D)=\sup\{t\in\mathbb R_+| (X,tD)\) is \(F\)--pure at \(x\}\). The pair \((X, tD)\) is \(F\)--pure at \(x\) if the canonical inclusion \(\mathcal O_{X,x}\to\mathcal O_X(t(q-1)D)_x^{\frac{1}{q}}\) splits as an \(\mathcal O_{X,x}\)--module homomorphism for all powers \(q\) of \(p\). It is proved that the \(F\)--pure thresholds of curve singularities on a smooth surface are rational. A lower bound for \(F\)--pure thresholds in the homogeneous case is given. Based on an appendix by P. Monsky the rationality and discreteness of \(F\)--jumping exponents, the smallest positive one of which is the \(F\)--pure threshold, is proved.
scientific article; zbMATH DE number 5121794