2019/01/23 by Cluckers, Raf, Mustata, Mircea
#11L07 #14B05 #14E18 #14J17 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1901.08111
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 lct(f, Jf2)>1 if and only if the hypersurface defined by f has rational singularities. Moreover, if this is not the case, then lct(f, Jf2)=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, Jf2). 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).