Jet schemes, log discrepancies and inversion of adjunction

By L. Ein, M. Mustaţă, Takehiko Yasuda. 2002

96 citations recorded by the source corpus.

Singularities play a key role in the Minimal Model Program. In this paper we show how some of the open problems in this area can be approached using jet schemes. Let (X, Y ) be a pair, where X is a Q-Gorenstein normal variety, and Y stands for a formal combination ∑k i=1 qi · Yi, where qi ∈ R+ and Yi ⊂ X are proper closed subschemes. Fix a closed subset ∅ 6 = W ⊆ X . Using a suitable resolution of singularities for the pair (X, Y ) one can define numerical invariants mld(W ;X, Y ), called minimal log discrepancies. These invariants in turn can be used to define the classes of singularities which appear in Mori Theory. We provide a way to compute minimal log discrepancies using arcs and jets. The mth jet scheme Xm of X is given set-theoretically as Hom(SpecC[t]/(t), X). The limit of these schemes is the space of arcs X∞ = Hom(SpecC[[t]], X). This is an infinite dimensional space, but we may associate to (X, Y ) a family of subsets of X∞ of finite codimension. Given W , if we restrict these subsets over W , then from their codimensions we can compute mld(W ;X, Y ). We stress that this characterization holds in complete generality. It extends the results in [Mu1], [Ya], and [ELM], where criteria were given for having nonnegative log discrepancy, under certain hypotheses on the singularities of X . The main ingredient in the proof of this characterization is the theory of motivic integration on singular varieties, developed by Denef and Loeser in [DL1].

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fe672da966ea458f1b7cdb7be778e9c1b3d283c2

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. S. Ishii
2. M. Mustaţă
3. Chi Li
4. Mattias Jonsson
5. H. Blum
6. Yuchen Liu
7. Kiumars Kaveh
8. Askold Khovanskii

In Geometry and Topology

1. S. Ishii
2. Yusuke Nakamura
3. M. Mustaţă
4. Tommaso de Fernex
5. Chi Li
6. L. Ein
7. H. Blum
8. Mattias Jonsson

In Mathematical Physics

1. S. Ishii
2. Yusuke Nakamura
3. M. Mustaţă
4. Tommaso de Fernex
5. L. Ein
6. Karl Schwede
7. H. Blum

Citations in the corpus

• The arc space of a toric variety (2003, background): “In fact, papers [5], [10], [11] by Mustaţǎ, Ein and Yasuda show that geometric properties of the jet schemes determine certain properties of the singularities of the base space.”
• Motivic integration over Deligne–Mumford stacks (2003, unknown): “This kind of integration over a singular variety was considered in [DL2] (see also [Loo]) in relation to the McKay correspondence and in [Yas2] and [EMY] in relation to the discrepancy of singularities.”
• Local and Global Methods in Algebraic Geometry (2015, result): “ Semicontinuity conjecture and the ACC conjecture) imply termination of flips. The Semicontinuity conjecture is believed to be the easier of the two problems. In fact, this is known in some cases (see [EMY03] for the case of smooth varieties and [Nak] for the case of varieties with quotient singulari”