Jet Schemes of Quasi-Ordinary Surface Singularities

Abstract

In this paper we give a complete description of the irreducible components of the jet schemes (with origin in the singular locus) of a two-dimensional quasi-ordinary hypersurface singularity. We associate with these components and with their codimensions and embedding dimensions, a weighted graph. We prove that the data of this weighted graph is equivalent to the data of the topological type of the singularity. We also determine a component of the jet schemes (or equivalently, a divisor on A 3 ), that computes the log canonical threshold of the singularity embedded in A 3 . This provides us with pairs X ⊂ A 3 whose log canonical thresholds are not contributed by monomial divisorial valuations. Note that for a pair C ⊂ A 2 , where C is a plane curve, the log canonical threshold is always contributed by a monomial divisorial valuation (in suitable coordinates of A 2 ). 1 , . . . , x 1 n d ]]). Moreover, some special exponents (called the characteristic exponents) which belong to the support of this series, are complete invariants of the topological type of the singularity (see [16]). In particular, they determine invariants which come from resolution of singularities, like the log canonical threshold or the Motivic zeta functions ([8], [3], [9]). They also give insights about the construction of a resolution of singularities ([6], [7], [18], [39]).Our aim is to construct some comparable complete invariants for all type of singularities. We search for such invariants in the jet schemes. For m ∈ N, the m-jet scheme, denoted by X m , is a scheme that parametrizes morphisms Spec C[t]/(t m+1 ) −→ X. Intuitively we can think of it as parametrizing arcs in an ambient space, which have a large contact, depending on m, with X. We know already that some invariants which come from resolution of singularities are encoded in jet schemes ([32], [13]).We want to extract from the jet schemes information about the singularity, which can be expressed in terms of invariants of resolutions of singularities. With this, our next goal is to construct a resolution of singularities by using invariants of jet schemes. For specific types of singularities, the knowledge of the irreducible components of the jet schemes X m of a singular variety X, together with some invariants of them, like dimension or embedding dimension, permits to determine deep invariants of the singularity of X: the topological type in the case of curves (see [26]), and the analytical type in the case of normal toric surfaces (see [27] and [28]). Moreover, in the case of irreducible plane curves, the minimal embedded resolution can be constructed from the jet schemes ([23]), and the same for rational double point singularities ([31]). Notice that, understanding the structure of jet schemes for particular singularities, remains a difficult problem. These structures have been studied in [40] and [12] for determinantal varieties, in [26] for plane curve singularities, [27] and [28] for normal toric surfaces, in [29] for rational double point surface singularities, and in [3...

On the embedded Nash problem

“…In general, the passage from a numerical characterization of the configuration of irreducible components of the jet schemes of D to a geometric characterization (i.e. in terms of valuations) of the irreducible components of the contact loci of (X, D) is more difficult, and only partial solutions to the embedded Nash problem are obtained, see [Mo14,Mo17,MoPl18,K+20,CM21,Ko22].…”

The embedded Nash problem of birational models of rational triple singularities

“…6,5 (C 5,1 ), we compute F 6 modulo the ideal (x 0 , y 0 , y 1 , z 0 , z 1 ) and we find C 6,1 := π −1 6,5 (C 5,1 ) = V (x 0 , y 0 , y 1 , z 0 , z 1 , z 3 2 + x 2 1 y 2 2 ) ⊂ C 3 6 . Notice that C 6,1 is isomorphic to the product of an affine space and the hypersurface defined by {z 3 2 + x 2 1 y 2 2 = 0}; this hypersurface is a Hirzebruch-Jung singularity which is well known to be an irreducible quasi-ordinary singularity [4]; in particular C 6,1 is irreducible. Actually, we will see that C 6,1 will give rise to an irreducible component of X Sing 6 whose weight vector is same as the weight vector associated with C 5,1 , so it is not an essential component (see definition above): the divisorial valuation associated with it is not monomial while a divisorial valuation associated with an essential component is monomial.…”

“…Proposition 4.1. For an E 6,0 -singularity, the monomial valuations associated with the vectors (0, 1, 1), (0, 2, 1), (1, 1, 1), (0, 3, 2), (1, 1, 2), (1, 2, 2), (2, 1, 2), (2, 1, 3), (2, 2, 3), (3, 2, 3), (3, 2, 4), (3,3,4), (4,3,5), (5,4,6) belong to EV (X).…”

“…Let X be a surface of type E 0,7 . The monomial valuations associated with the vectors {(0, 1, 1), (0, 2, 1), (1, 1, 1), (0, 3, 2), (1, 1, 2), (1, 2, 2), (2, 1, 2), (2,2,3), (3,2,3), (3,2,4), (3,3,4), (4,3,5), (5,3,5)) (5,4,6), (6,4,7), (7,5,8), (9,6,10)} belong to EV (X). There exists a toric birational map µ Σ : Z Σ −→ C 3 which is an embedded resolution of X ⊂ C 3 such that the irreducible components of the exceptional divisor of µ Σ correspond to the irreducible components of the m-th jet schemes of X (centered at the singular locus and the intersection of X with the coordinate hyperplane).…”

Resolving singularities of curves with one toric morphism

“…One can also detect the essential valuations on a graph which is associated with the jet schemes of the curve singularity [29,32]. This latter graph makes sense also for higher dimensional singularities [5] and the loc.cit. program suggests that one can detect essential valuations on it [34].…”