Bounds for log canonical thresholds with applications to birational rigidity

By Tommaso de Fernex, L. Ein, M. Mustaţă. 2002

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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\leq N\leq 12.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/15d8b265be4981c3ffbda1593bb92c412e9120a0

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Citations in the corpus

IMPANGA lecture notes on log canonical thresholds (2011, background): “ for hypersurfaces of degree n in Pn). This is a point of view due to Corti [Cor] on the classical approach to the non-rationality of a quartic threefold of Iskovskikh and Manin [IM]. See for example [dFEM4] for an application of this point of view. The present notes are based on a mini-course I gav”
Log Canonical Thresholds of Del Pezzo Surfaces (2007, unknown): “Moreover, if X = Cn and D = (φ = 0), then the equality lctO ( X,D ) = c0 ( φ ) holds (see [10]).”
Existence of valuations with smallest normalized volume (2016, background): “)ne(m) lct(a)ne(a) where m x is the maximal ideal of O X;x. The authors of the previous theorem were motivated by their interest in singularity theory, as well as applications to birational rigidity [dFEM03] [dFEM04] [dF013]. Li’s interest in volume minimization stems from questions concerning K-sem”
The rationality problem for fields of invariants under linear algebraic groups (with special regards to the Brauer group) (2005, background): “It has witnessed a strong development: birational rigidity and birational super-rigidity, work of Iskovskikh [56], Pukhlikov [112,113], de Fernex-Ein-Mustaţă [38].”
Threefold thresholds (2003, background): “[22] proceeds by analysing all possible exceptional divisors on a smooth surface that have log discrepancy zero with respect to some divisor KX + ∆.”