Jet schemes of rational double point singularities

Abstract

We prove that for m ∈ N, m big enough, the number of irreducble components of the schemes of m−jets centered at a point which is a double point singularity is independent of m and is equal to the number of exceptional curves on the minimal resolution of the singularity. We also relate some irreducible components of the jet schemes of an E 6 singularity to its "minimal" embedded resolutions of singularities.

Jet schemes of toric surfaces

“…For m ∈ N, m 1, we determine the irreducible components of the m-th jet scheme S m of a toric surface S. We give formulas for their number and their dimensions in terms of m, and invariants of the cone defining S. For a given m, we classify these irreducible components by an integer invariant that we call index of speciality. We prove that for m big enough, the components with index of speciality 1, are in 1-1 correspondence with the exceptional divisors that appear on the minimal resolution of S. This is to compare with a result that we have obtained in [12] for rational double point singularities.…”

“…We suppose that e > 3, the case e = 3, i.e. the rational double point S = A c 2 −1 is studied in [12]. Analyzing the convex hull of σ ∨ ∩ M, where M is the dual lattice of N, Riemenschneider has exhibited the generators of the ideal defining S in [16]; these are:…”

Jet schemes and generating sequences of divisorial valuations in dimension two

“…This follows from the definition of ν E in terms of jet schemes. Indeed, C p−1 gives rise to an irreducible component W of Cont p (C) (see the discussion after theorem 3.2 in [Mo2]), and we have that v W (h) = min γ∈W {ord t γ * (h)},…”

Jet schemes and minimal toric embedded resolutions of rational double point singularities

“…These information will be used in the next section to obtain the canonical resolution of these singularities. We will all the cases of simple singularities (see their defining equations below); the case of E 6 (see below the defining equation) has been treated in [Mo1], but we consider it briefly here for the convenience of the reader.…”

“…and is defined by the ideal I m = (F 0 , F 1 , ..., F m ). From Section 3 in [Mo4] and section 3.2 in [Mo1], we are able to determine the set EC(X). This is the subject of the following lemma.…”