Shihoko Ishii
This paper shows how properties of jet schemes relate to those of the singularity on the base scheme. We will see that the jet scheme's properties of being Q-factorial, Q-Gorenstein, canonical, terminal and so on are inherited by the base scheme.
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We study logarithmic jet schemes of a log scheme and generalize a theorem of M. Mustata from the case of ordinary jet schemes to the logarithmic case. If X is a normal local complete intersection log variety, then X has canonical singularities if and only if the log jet schemes of X are irreducible.
If two schemes are isomorphic, then their $m$-jet schemes are isomorphic for all $m$. In this paper we consider the converse problem. We prove that if an isomorphism of the $m$-jet schemes is induced from a morphism of the base schemes, then the morphism of the base schemes is an isomorphism. But we also prove that just the existence of isomorphisms between $m$-jet schemes does not yield the existence of an isomorphism between the base schemes.
Smoothness and jet schemes
本文给出了代数闭域上有限型概形格式非奇异的喷流格式的一些判别准则。对于特征为零的基场，该格式是非奇异的当且仅当其喷流格式的截断态射之一是平坦的。对于正特征情形，在约化条件下得到了类似的刻画。我们还通过简单的讨论得到了该格式非奇异的充要条件是它的一个喷流格式是非奇异的。
Given a scheme X over a field k, a generalized jet scheme parametrizes maps from Spec(A) to X, where A is a finite-dimensional, local algebra over k. We give an overview of known results concerning the dimensions of these schemes when A has embedding dimension one, when they are related to invariants of singularities in birational geometry. We end with a discussion of more general jet schemes.
Jet schemes, arc spaces and the Nash problem
本文介绍了代数簇的喷流格式和弧空间。我们还介绍了弧族上的纳什问题。
For $m\in \IN, m\geq 1,$ we determine the irreducible components of the $m-th$ jet scheme of a toric surface $S.$ For $m$ big enough, we connect the number of a class of these irreducible components to the number of exceptional divisors on the minimal resolution of $S.$
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 $C$. This structure of the Jet Schemes determines and is determined by the topological type of $C$.
Dimensions of jet schemes of log singularities
本文利用 jet 概型的维数刻画 Kawamata 对数终端奇点和对数典范奇点，推广了 Mustata 的已有结果。
Jet Schemes, Newton Polygons and Continued Fractions
本研究针对Newton非退化平面曲线奇点的jet方案，识别出可由第一象限格点游走构造的jet分量子图，确定了jet方案所有不可约分量，证明子图编码曲线奇点嵌入拓扑类型，还定义并证明了相关生成级数的有理性及明确极点。
Jet Schemes of Locally Complete Intersection Canonical Singularities
我们证明了如果X是局部完备的交簇，则X有所有的jet格式不可约当且仅当X有正则奇点。在将X嵌入到一个光滑簇Y中之后，我们利用运动积分来表示X具有不可约喷流格式的条件，该条件是由X在Y中的嵌入分解得到的。我们证明了这个条件与正则奇点是等价的。在附录中，这个结果被用来证明一个推广的Kostant的自由度定理设置喷流计划。
Using the structure of the jet schemes of rational double point singularities, we construct "minimal embedded toric resolutions" of these singularities. We also establish, for these singularities, a correspondence between a natural class of irreducible components of the jet schemes centered at the singular locus and the set of divisors which appear on every "minimal embedded toric resolution". We prove that this correspondence is bijective except for the $E_8$ singulartiy. This can be thought as an embedded Nash correspondence for rational double point singularities.
Kempf proved that the theta divisor of a smooth projective curve C has rational singularities. In this paper we estimate the dimensions of the jet schemes of the theta divisor and show that all these schemes are irreducible. In particular, we recover Kempf's theorem in this way. For general projective smooth curves, our method also gives a formula for the log canonical threshold of the pair (\pic^d(C), W^r_d(C)).
Given an arbitrary projective birational morphism of varieties, we provide a natural and explicit way of constructing relative compactifications of the maps induced on the main components of the jet schemes. In the case the morphism is the Nash blow-up of a variety, such relative compactifications are shown to be given by the Nash blow-ups of the main components of the jet schemes.
Jet schemes and singularities
我们给出了一个独立的介绍奇异品种的喷气计划的基本结果。应用程序给出了不变量的奇异性，如最小的日志差异。我们简化了我们的老办法反演的附加局部完整的交叉品种，我们证明了一个更一般的结果，任意Q-Gorenstein品种嵌入在非奇簇（这一声明已独立证明也由Kawakita）。
Using the theory of jet schemes, we give a new approach to the description of a minimal generating sequence of a divisorial valuations on $\textbf{A}^2.$ For this purpose, we show how one can recover the approximate roots of an analytically irreducible plane curve from the equations of its jet schemes. As an application, for a given divisorial valuation $v$ centered at the origin of $\textbf{A}^2,$ we construct an algebraic embedding $\textbf{A}^2\hookrightarrow \textbf{A}^N,N\geq 2$ such that $v$ is the trace of a monomial valuation on $\textbf{A}^N.$ We explain how results in this direction …
This paper summarizes the results at the present moment about singularities with respect to the Mather-Jacobian log discrepancies over algebraically closed field of arbitrary characteristic. The basic point is the Inversion of Adjunction with respect to Mather-Jacobian discrepancies holds in arbitrary characteristic. Based on this fact we will reduce many geometric properties of the singularities into the problem on jet schemes and try to avoid discussions which are distinctive for characteristic 0.
Hilbert series of second order jets of determinantal varieties
本文针对行列式簇射流概型中希尔伯特函数难以计算的问题，成功给出了 $2\ imes n$ 矩阵极大子式情形下二阶射流概型的希尔伯特函数。
Singularities of Pairs via Jet Schemes
设X是光滑簇，Y是X的闭子概形.通过比较X上的运动积分和（X，Y）的对数分辨率上的运动积分，证明了（X，Y）的对数正则阈值c（X，Y）=dim X-sup_m{（dim Y_m}/（m+1）}，其中Y_m是Y的第m个喷流格式.我们展示了如何使用这个公式来研究对数正则阈值。特别地，我们给出了Demailly和Kollár的半连续性定理的证明。
Motivated by Demailly's strategy towards the Kobayashi hyperbolicity conjecture, we study the action on the k-jets of germs of holomorphic discs in a complex manifold X of the reparametrization group of k-jets of germs of biholomorphisms of the source. This reparametrization group is a subgroup of the general linear group GL(k) which is not reductive, but nonetheless we show that its invariants for any linear action which extends to GL(k) form a finitely generated algebra, and give a new geometric description of the Demailly-Semple algebra of invariant jet differentials.
Multiplicity of jet schemes of monomial schemes
本文研究单项格式的喷流格式。已知它们是等维的，但通常不约化。因此，我们进一步研究它们的结构，给出了一个公式，沿每个组件的喷流计划的一般约化单项超曲面（即，一个简单的正规交叉因子的情况下）的重数。
On jet schemes of determinantal varieties
本文利用jet scheme方法研究行列式簇，发现了其与jet schemes间Hilbert级数的新联系，证明了特定情形下定义理想的可壳性，并为一般情形提供了定义不可约分支的多项式族。
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 $\mathbb{A}^3$), that computes the log canonical threshold of the singularity embedded in $\mathbb{A}^3$.…
The theory of relative logarithmic jet spaces is developed for log schemes. With this theory the existence of bounds of intersection multiplicities of curves and divisors on certain log schemes is established. This result extends those of Noguchi and Winkelmann by replacing the semi-abelian condition by a differential one.
For $m\in \mathbb{N}, m\geq 1,$ we determine the irreducible components of the $m-th$ jet scheme of a normal toric surface $S.$ We give formulas for the number of these components and their dimensions. When $m$ varies, these components give rise to projective systems, to which we associate a weighted graph. We prove that the data of this graph is equivalent to the data of the analytical type of $S.$ Besides, we classify these irreducible components by an integer invariant that we call index of speciality. We prove that for $m$ large enough, the set of components with index of speciality $1,$ i…
This paper studies the singularities of jet schemes of homogeneous hypersurfaces of general type. We obtain the condition of the degree and the dimension for the singularities of the jet schemes to be of dense $F$-regular type. This provides us with examples of singular varieties whose $m$-jet schemes have rational singularities for every $m$.
Jet schemes of determinantal varieties
本文研究了行列式型喷流格式的格式结构。我们表明，在一般情况下，这些喷流计划不是不可约的。在秩至多为1的$r \times s$矩阵的行列式簇$X$的情况下，我们给出了它的喷流格式的每个分量的维数的公式。作为应用，我们计算了对数正则阈值对$(\mathbb{A}^{rs},X)$。
Jet Schemes of the Commuting Matrix Pairs Scheme
我们证明了对所有的$k\ge 1$，存在一个整数$N(k)$，使得对所有的$n\ge N(k)$，交换$n\times n$矩阵对上的$k$阶jet格式是可约的.在另一端的频谱，它是已知的，对于所有的$k\ge 1$，$k$ -第阶喷射计划的交换$2\times 2$矩阵是不可约的：我们表明，同样适用于$n=3$。
We prove that, for the jet scheme of a singular hypersurface, the blowup of a certain jet-related module is not an isomorphism. In conjunction with recent developments in the theory of Nash blowups, our result holds over fields of arbitrary characteristic. Our approach is based on explicit presentations given by a higher-order Jacobian matrix combined with a certain jet-related matrix.
One higher dimensional analog of jet schemes
我们发展了截断楔形格式的理论，这是一种高维的类似喷流格式的格式。我们证明了一些基本的性质，并给出了一个不可约准则截断楔形格式的局部完整的交叉品种类似Mustaeruginski的喷气计划。我们表明，减少的子计划结构的截断楔形计划的任何单项计划本身是一个单项计划在一个自然的方式，明确描述发电机的任何截断楔形计划的单项超曲面的每个最小素数。证明了约化单项式超曲面的截断楔格式的不可约分支都具有重数1。
In the present paper, we give a full description of the jet schemes of the polynomial ideal $\left( x_1\ldots x_n \right) \in k[x_1, \ldots, x_n]$ over a field of zero characteristic. We use this description to answer questions about products and intersections of ideals emerged recently in algorithmic studies of algebraic differential equations.
To each variety $X$ and a nonnegative integer $m$, there is a space $X_m$ over $X$, called the jet scheme of $X$ of order $m$, parametrizing $m$-th jets on $X$. Its fiber over a singular point of $X$ is called a singular fiber. For a surface with a rational double point, Mourtada gave a one-to-one correspondence between the irreducible components of the singular fiber of $X_m$ and the exceptional curves of the minimal resolution of $X$ for $m \gg 0$. In this paper, for a surface $X$ over complex number with a singularity of $A_n$ or $D_4$-type, we study the intersections of irreducible compone…
Derived jet and arc spaces
本文研究了导出代数几何中jet方案和arc空间的导出构造，展示了在光滑基方案或更一般的局部完全交log canonical奇点情况下，导出构造与经典版本一致，并为Mustață的定理提供了导出解释，同时对于更奇异的空间，得到了新的奇点不变量，形式为更高的同调群，并研究了导出jet和arc空间的cotangent复形，推广了经典jet和arc空间的微分sheaf公式，并得到了多个应用。
We study in this paper the jet schemes of the closure of nilpotent orbits in a finite-dimensional complex reductive Lie algebra. For the nilpotent cone, which is the closure of the regular nilpotent orbit, all the jet schemes are irreducible. This was first observed by Eisenbud and Frenkel, and follows from a strong result of Mustautca (2001). Using induction and restriction of "little" nilpotent orbits in reductive Lie algebras, we show that for a large number of nilpotent orbits, the jet schemes of their closure are reducible. As a consequence, we obtain certain geometrical properties of the…
Jet schemes, log discrepancies and Inversion of Adjunction
我们利用奇异空间的motivic积分理论，给出了弧空间的某些子集的余维的最小对数离散的一个特征。这是针对任意对$(X,Y)$进行的，其中$X$为正常值，Q为Gorenstein。作为第一个应用程序，我们证明了一个精确的版本的逆的附加，在环境变化是光滑的情况下。另一个应用是关于光滑簇最小对数偏差的连续性。
Jet schemes of singular surfaces of types $D_4^0$ and $D_4^1$ in characteristic $2$
本文研究了特征 $2$ 域上 $D_4$ 型二维有理双点 jet schemes 的奇异纤维，给出了 $D_4^0$ 和 $D_4^1$ 型奇点奇异纤维的不可约分解及分量构型。
We show in this paper that the principal component of the first order jet scheme over the classical determinantal variety of m x n matrices of rank at most 1 is arithmetically Cohen-Macaulay, by showing that an associated Stanley-Reisner simplicial complex is shellable.
In this article, given a scheme $X$ we show the existence of canonical lifts of Frobenius maps in an inverse system of schemes obtained from the fiber product of the canonical prolongation sequence of arithmetic jet spaces $J^*X$ and a prolongation sequence $S^*$ over the scheme $X$. As a consequence, for any smooth group scheme $E$, if $N^n$ denote the kernel of the canonical projection map of the $n$-th jet space $J^nE \rightarrow E$, then the inverse system $\{N^n\}_n$ is a prolongation sequence.
In this paper we explore the connection between asymptotic base loci and Newton-Okounkov bodies associated to infinitesimal flags. Analogously to the surface case, we obtain complete characterizations of augmented and restricted base loci. Interestingly enough, an integral part of the argument is a study of the relationship between certain simplices contained in Newton-Okoukov bodies and jet separation; our results also lead to a convex geometric description of moving Seshadri constants.
If a morphism of germs of schemes induces isomorphisms of all local jet schemes, does it follow that the morphism is an isomorphism? This problem is called the local isomorphism problem. In this paper, we use jet schemes to introduce various closure operations among ideals and relate them to the local isomorphism problem. This approach leads to a partial solution of the local isomorphism problem, which is shown to have a negative answer in general and a positive one in several situations of geometric interest.
We compute the minimal log discrepancies of determinantal varieties of square matrices, and more generally of pairs $\bigl(D^k,\sum α_i D^{k_i}\bigr)$ consisting of a determinantal variety (of square matrices) and an $\mathbb R$-linear sum of determinantal subvarieties. Our result implies the semicontinuity conjecture for minimal log discrepancies of such pairs. For these computations, we use the description of minimal log discrepancies via codimensions of cylinders in the space of jets; this necessitates the computations of an explicit generator for the canonical differential forms and the Na…
Kernels of Arithmetic Jet Spaces and Frobenius Morphism
本文研究了算术jet空间的核与Frobenius映射，揭示了其与Witt向量结构的联系。
Let $Z \subset \mathbb{A}^k$ be an affine scheme over $\C$ and $\J Z$ its jet scheme. It is well-known that $\mathbb{C}[\J Z]$, the coordinate ring of $\J Z$, has the structure of a commutative vertex algebra. This paper develops the orbifold theory for $\mathbb{C}[\J Z]$. A finite-order linear automorphism $g$ of $Z$ acts by vertex algebra automorphisms on $\mathbb{C}[\J Z]$. We show that $\mathbb{C}[\J^g Z]$, where $\J^g Z$ is the scheme of $g$--twisted jets has the structure of a $g$-twisted $\mathbb{C}[\J Z]$ module. We consider spaces of orbifold coinvariants valued in the modules $\mathb…
We consider the affine variety ${\mathcal{Z}_{2,2}^{m,n}}$ (or just "$Y$") of first order jets over ${\mathcal{Z}_{2}^{m,n}}$ (or just "$X$"), where $X$ is the classical determinantal variety given by the vanishing of all $2\times 2$ minors of a generic $m\times n$ matrix. When $2 < m \le n$, this jet scheme $Y$ has two irreducible components: a trivial component, isomorphic to an affine space, and a nontrivial component that is the closure of the jets supported over the smooth locus of $X$. This second component is referred to as the principal component of $Y$; it is, in fact, a cone and can …
Let $G$ be a complex reductive group and $V$ a $G$-module. Then the $m$th jet scheme $G_m$ acts on the $m$th jet scheme $V_m$ for all $m\geq 0$. We are interested in the invariant ring $\mathcal{O}(V_m)^{G_m}$ and whether the map $p_m^*\colon\mathcal{O}((V//G)_m) \rightarrow \mathcal{O}(V_m)^{G_m}$ induced by the categorical quotient map $p\colon V\rightarrow V//G$ is an isomorphism, surjective, or neither. Using Luna's slice theorem, we give criteria for $p_m^*$ to be an isomorphism for all $m$, and we prove this when $G=SL_n$, $GL_n$, $SO_n$, or $Sp_{2n}$ and $V$ is a sum of copies of the st…
Using Buium's theory of arithmetic differential characters, we construct a filtered $F$-isocrystal ${\bf H}(A)_K$ associated to an abelian scheme $A$ over a $p$-adically complete discrete valuation ring with perfect residue field. As a filtered vector space, ${\bf H}(A)_K$ admits a natural map to the usual de Rham cohomology of $A$, but the Frobenius operator comes from arithmetic differential theory and is not the same as the usual crystalline one. When $A$ is an elliptic curve, we show that ${\bf H}(A)_K$ has a natural integral model ${\bf H}(A)$, which implies an integral refinement of a re…
The jet schemes of the nilpotent cone of $\mathfrak{gl}_n$ over $\mathbb{F}_\ell$ and analytic properties of the Chevalley map
本文研究了正特征下$\mathfrak{gl}_n$nilpotent cone的jet方案维数界，证明了Chevalley映射的解析性质，尤其在特征大于n/2时无条件成立，同时指出零特征情况与Harish-Chandra积分可积定理相关。
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