An invariant detecting rational singularities via the log canonical threshold

Abstract

We show that if f is a nonzero, noninvertible function on a smooth complex variety X and J_f is the Jacobian ideal of f, then lct(f, J_f^2)>1 if and only if the hypersurface defined by f has rational singularities. Moreover, if this is not the case, then lct(f, J_f^2)=lct(f). We give two proofs, one relying on arc spaces and one that shows that the minimal exponent of f is at least as large as lct(f, J_f^2). In the case of a polynomial over the algebraic closure of Q, we also prove an analogue of this latter inequality, with the minimal exponent replaced by the motivic oscillation index moi(f).