2022/02/17 by Cluckers, Raf, Kollár, János, Mustaţă, Mircea · 1 citation
#11L07 #14B05 #14E18 #14J17 #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2202.08425
We show that if f is a nonzero, noninvertible function on a smooth complex variety X and Jf is the Jacobian ideal of f, then \rm lct(f,Jf2)>1 if and only if the hypersurface defined by f has rational singularities. Moreover, if it does not have rational singularities, then \rm lct(f,Jf2)=\rm lct(f). We give two proofs, one relying on arc spaces and one that goes through the inequality \widetildeα(f)≥\rm lct(f,Jf2), where \widetildeα(f) is the minimal exponent of f. In the case of a polynomial over Q, we also prove an analogue of this latter inequality, with \widetildeα(f) replaced by the motivic oscillation index \rm moi(f). We also show a part of Igusa's strong monodromy conjecture, for poles larger than -\rm lct(f,Jf2). We end with a discussion of lct-maximal ideals: these are ideals I with the property that \rm lct(I)<\rm lct(J) for every J with I\subsetneq J.