Bounds for log canonical thresholds with applications to birational rigidity

2002/12/31 by Tommaso de Fernex, Lawrence Ein, Mircea Mustata · 2 citations
Mathematics · #math.AG #msc:14B05 #msc:14C17 #msc:14E05

paper · pdf

published as Math. Res. Lett. 10 (2003), 219-236. · 16 pages, AMS-LaTeX; v2: corrected reference; v3: last application, to the complete intersection of type (2,6) in P^8, was removed due to a numerical error; all other results are unchanged; final version, to appear in Math. Res. Lett

arxiv created 2003/02/14 · arxiv updated 2009/11/30

Abstract

We use intersection theory, degeneration techniques and jet schemes to study log canonical thresholds. Our first result gives a lower bound for the log canonical threshold of a pair in terms of the log canonical threshold of the image by a suitable smooth morphism. This in turn is based on an inequality relating the log canonical threshold and the Samuel multiplicity, generalizing our previous result from math.AG/0205171. We then give a lower bound for the log canonical threshold of an affine scheme defined by homogeneous equations of the same degree in terms of the dimension of the non log terminal locus (this part supersedes math.AG/0105113). As an application of our results, we prove the birational superrigidity of every smooth hypersurface of degree N in P^N, if 4≤ N≤ 12.

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