\(F\)-jumping and \(F\)-Jacobian ideals for hypersurfaces (Q886982)

The authors introduce \(F\)-jumping and \(F\)-Jacobian ideals for hypersurfaces in positive characteristic to study their singularities. Both are defined using the \(D\)-module \(M_\alpha\) introduced by Blickle, Mustaţă and Smith [\textit{M. Blickle} et al., Trans. Am. Math. Soc. 361, No. 12, 6549--6565 (2009; Zbl 1193.13003)]. For example, The \(F\)-jumping ideal \(J(f^\alpha)\) is defined as the ideal in \(R\) such that \(J(f^\alpha)e_\alpha= Re_\alpha \cap N_\alpha\) where \(N_\alpha\) is the unique simple nonzero \(D\)-submodule of \(M_\alpha\) and \(e_\alpha\) is the generator of \(M_\alpha\) as an \(R_f\)-module. The main result is a characterization of the \(F\)-jumping numbers for hypersurfaces. Let \(R\) be an \(F\)-finite regular domain of characteristic \(p>0\) and let \(f\in R\) be a nonzero element. Let \(\alpha\in\mathbb{Q}\) be a rational number with denominator not divisible by \(p\). Then \(\alpha\) is not an \(F\)-jumping number if and only if \(M_\alpha\) is a simple \(D\)-module (equivalent to being simple as an \(F^e\)-module), and also if and only if the \(F\)-jumping ideal \(J(f^\alpha)\) is the unit ideal.