相关论文: Singularities of log varieties via jet schemes

For $m \in \mathbb{N}$, we determine the irreducible components of the $m$-th Jet Scheme of a complex branch $C$ and give formulas for their number $N(m)$ and for their codimensions, in terms of $m$ and the generators of the semigroup of…

We develop the theory of truncated wedge schemes, a higher dimensional analog of jet schemes. We prove some basic properties and give an irreducibility criterion for truncated wedge schemes of a locally complete intersection variety…

The singularity structure of solutions of a class of Hamiltonian systems of ordinary differential equations in two dependent variables is studied. It is shown that for any solution, all movable singularities, obtained by analytic…

These lecture notes provide a unified overview of most known canonical desingularization methods in characteristic zero. It starts with discussing the classical method, and then proceeds with the recently discovered ones: logarithmic…

We prove that a Cohen-Macaulay normal variety $X$ has Du Bois singularities if and only if $\pi_*\omega_{X'}(G) \simeq \omega_X$ for a log resolution $\pi: X' \to X$, where $G$ is the reduced exceptional divisor of $\pi$. Many basic…

M. Mustata used jet schemes to compute the multiplier ideals of reduced hyperplane arrangements. We give an alternate proof using a log resolution, which is simpler and allows us to consider non-reduced arrangements. By applying the idea of…

We prove the existence of log canonical modifications for a log pair. As an application, together with Koll\"ar's gluing theory, we remove the assumption in the first named author's work [Odaka11], which shows that K-semistable polarized…

We introduce the jet schemes of a holomorphic foliation, and thereby prove an alternate characterisation of simple singularities of codimension-$1$ foliations, independent of any normal form. This leads to an equivalent condition for the…

The minimal log discrepancy is an invariant of singularities that plays an important role in the birational classification of algebraic varieties. Shokurov conjectured that the minimal log discrepancy can always be bounded from above in…

This work provides a unified formalism for studying difference and (Hasse-) differential algebraic geometry, by introducing a theory of "iterative Hasse rings and schemes". As an application, Hasse jet spaces are constructed generally,…

We shall investigate index 1 covers of 2-dimensional log terminal singularities. The main result is that the index 1 cover is canonical if the characteristic of the base field is different from 2 or 3. We also give some counterexamples in…

We give a sufficient condition under which the moduli space of morphisms between logarithmic schemes is quasifinite under the moduli space of morphisms between the underlying schemes. This implies that the moduli space of stable maps from…

As is well known, the "usual discrepancy" is defined for a normal Q-Gorenstein variety. By using this discrepancy we can define a canonical singularity and a log canonical singularity. In the same way, by using a new notion, Mather-Jacobian…

We describe the generic singularity of a Schubert variety of type A on each irreducible component of its singular locus. This singularity is given either by a cone of rank one matrices, or a quadratic cone.

We propose a log-concavity conjecture for BPS invariants arising in the enumerative geometry of planar curve singularities, identified with the local Euler obstructions of Severi strata in their versal deformations. We further extend this…

We survey some recent topics on singularities, with a focus on their connection to the minimal model program. This includes the construction and properties of dual complexes, the proof of the ACC conjecture for log canonical thresholds and…

This paper obtains criteria for a Fano variety X with normal crossing singularities defined over an algebraically closed field of characteristic zero, to be smoothable. The difference with the original version is that the theory of…

We prove that the base space of a log smooth family of log canonical pairs of log general type is of log general type as well as algebraically degenerate, when the family admits a relative good minimal model over a Zariski open subset of…

We prove that the class of log canonical rational singularities is closed under the basic operations of the minimal model program. We also give some supplementary results on the minimal model program for log canonical surfaces.

We study the problem of resolving singularities via the blow-up of the module of derivations. Our main results are a positive answer for the case of curves and log-canonical surface singularities, i.e., a finite sequence of blow-ups along…