2001/05/14 by Lawrence Ein, Ein, Lawrence, Mircea Mustata +1
Mathematics · #14Bo5 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14Bo5
paper · pdf · doi:10.48550/arxiv.math/0105113
6 pages. We generalize the lower bound and the characterization for equality to the case when the hypersurface has a singular locus of arbitrary (but fixed) dimension
arxiv created 2001/05/31 · arxiv updated 2009/11/30
We prove that if Y is a hypersurface of degree d in Pn with isolated singularities, then the log canonical threshold of (Pn,Y) is at least minn/d,1. Moreover, if d is at least n+1, then we have equality if and only if Y is the projective cone over a (smooth) hypersurface in Pn-1. In the case when Y is a hyperplane section of a smooth hypersurface in Pn+1, Cheltsov and Park have proved that Y has isolated singularities and they have obtained the above lower bound for the log canonical threshold. Moreover they made the conjecture about the equality case (for d=n+1) and they proved that the conjecture follows from the Log Minimal Model Program. The purpose of this note is to give an easy proof of their conjecture using the description of the log canonical threshold in terms of jet schemes.