Enrique Chavez Martinez, Daniel Duarte, Arturo Giles Flores
We introduce a higher-order version of the tangent map of a morphism and find a matrix representation. We then apply this matrix to solve a conjecture by T. Yasuda regarding the semigroup of the higher Nash blowup of formal curves. We first show that the conjecture is true for toric curves. We conclude by exhibiting a family of non-monomial curves where the conjecture fails.
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Nash blowups of toric varieties in prime characteristic
本文研究素特征域上Nash吹胀的奇点消解性质，证明迭代规范化Nash吹胀可消解正规环面曲面奇点，并引入对数Jacobian理想的素特征版本以实现对任意维环面簇Nash吹胀的组合描述。
On the zero locus of ideals defining the Nash blowup of toric surfaces
我们考虑的问题，找到一个理想的爆破定义的纳什爆破的复曲面，使其零轨迹与奇异集的复曲面。
The higher Nash blowup of an algebraic variety replaces singular points with limits of certain spaces carrying higher order data associated to the variety at non-singular points. In the case of normal toric varieties we give a combinatorial description of the higher Nash blowup in terms of a Gröbner fan. This description will allow us to prove the analogue of Nobile's theorem on the usual Nash blowup in this context. More precisely, we prove that for a normal toric variety, the higher Nash blowup is an isomorphism if and only if the variety is non-singular.
Classification of normal toric surfaces resolved by a single Nash blowup
本文通过连分数方法对由单次Nash吹胀解析的正常尖点曲面进行了完整分类。
Higher Nash blowups
本文引入高阶纳什爆破概念，详细研究曲线情形，并证明特征零下任何曲线都可由足够高阶的纳什爆破实现去奇异化。
Computational aspects of the higher Nash blowup
代数簇的高阶Nash爆破将奇点替换为某些空间的极限，这些空间携带与该代数簇在非奇点处相关联的高阶数据。在本文中，我们将定义一个高阶雅可比矩阵，它将允许我们对超曲面的高阶Nash爆破进行显式计算。首先，我们将推广一个已知的方法来计算这种修改的纤维。其次，我们将给出一个明确的描述理想的爆破给出了更高的纳什爆破。作为结果，我们将推导出正规超曲面的Nynth定理的高阶版本。
Nash blowup fails to resolve singularities in dimensions four and higher
本文证明在任意特征的代数闭域上，对四维及以上代数簇迭代 Nash blowup 或规范化 Nash blowup 均不能消解奇点。
Characteristic-free normalized Nash blowup of toric varieties
本文研究环簇的标准化 Nash 爆破，引入锥体条件使定义多面体不依赖基域特征，从而推导奇点消解性质，恢复正特征已知结果并给出任意特征下非奇异的新族。
Non-normalized Nash blowup fails to resolve singularities in dimension three
本文证明了在特征零域上，迭代非正规化的Nash吹胀未必能消解三维代数簇的奇点。
We prove that the higher Nash blowup of a normal toric variety defined over a field of positive characteristic is an isomorphism if and only if it is non-singular. We also extend a result by R. Toh-Yama which shows that higher Nash blowups do not give a one-step resolution of the $A_3$-singularity. These results were previously known only in characteristic zero.
Flag higher Nash blowups
本文引入另一种更高阶纳什爆破，证明其与乘积和光滑态射相容，并证明曲线乘积可通过两种版本实现去奇异化。
On a Computational Approach to the Nash Blowup Problem
本文通过计算方法发现纳什吹胀猜想的反例，并提供了无法通过标准化纳什吹胀解析的典范簇例子，同时展示了大量计算证据支持猜想的剩余开放案例。
Nash blowups of normal toric surfaces: the case of one and two segments
本文研究正规环面曲面奇点的Nash爆破消解问题，证明当对应半群的极小生成集位于一段或两段时迭代Nash爆破可消解奇点，并给出任意多段情形成立的例子。
Nash blowups of 2-generic determinantal varieties in positive characteristic
本文通过显式描述toric结构并证明Nash吹胀组合学无特征依赖，得出正特征域上2-generic行列式簇的Nash吹胀非奇异。
We show that the $ n $-th Nash blowup of the toric surface singularity of type $ A_3 $ is singular for any $ n > 0 $. It was known that the normalization of the $ n $-th Nash blowup of a toric variety is also a toric variety associated to the Gröbner fan of a certain ideal $ J_n $. In our case, we prove that the Gröbner fan contains a non-regular cone. We determine minimal generators of the initial ideal of $ J_n $ with respect to a certain monomial ordering, and show that the reduced Gröbner basis of $ J_n $ has polynomials of certain forms for each $ n $.
A one-step counterexample to the normalized Nash blowup conjecture
本文构造了一个五维的正常奇异仿射交错簇，其归一化纳什吹胀已包含与原簇同构的仿射开子簇，从而在五维及以上所有特征下给出单步反例。
Mather-Yau's type theorem for higher Nash blowup algebras
本文针对任意域上的超曲面奇点，证明其局部环同构类型由高阶Nash爆破代数完全决定，先证该代数在接触等价下的稳定性，再用Samuel经典结果证逆命题，推广了经典Mather-Yau定理。
We initiate the study of Nash blowups in prime characteristic. First, we show that a normal variety is non-singular if and only if its Nash blowup is an isomorphism, extending a theorem by A. Nobile. We also study higher Nash blowups, as defined by T. Yasuda. Specifically, we give a characteristic-free proof of a higher version of Nobile's Theorem for quotient varieties and hypersurfaces. We also prove a weaker version for $F$-pure varieties.
Non-existence of negative derivations on the higher Nash blowup local algebra
本文证明具有孤立奇点的加权齐次多项式的高次Nash爆破局部代数在$n\geq2$时无负加权导子，肯定回答了Hussain-Ma-Yau-Zuo猜想。
Bilipschitz geometry of real surface singularities whose tangent cone is a plane
本文研究实曲面奇点在双Lipschitz等价下Nash锥与例外射线的行为，针对切锥为平面的孤立奇点情形。
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.
Our previous papers introduce topological notions of normal crossings symplectic divisor and variety, show that they are equivalent, in a suitable sense, to the corresponding geometric notions, and establish a topological smoothability criterion for normal crossings symplectic varieties. The present paper constructs a blowup, a complex line bundle, and a logarithmic tangent bundle naturally associated with a normal crossings symplectic divisor and determines the Chern class of the last bundle. These structures hav applications in constructions and analysis of various moduli spaces. As a coroll…
On Resolving Singularities
设V是特征为零的域k上的不可约仿射代数簇，设（f_0，...，f_m）是坐标环的元素序列。关于f_i及其导数，可能没有初等条件来决定V沿（f_0，...，f_m）是非奇异的。结果是，确实存在这样一个基本条件，涉及到$f_i,$的一阶和二阶导数，只要我们允许某些奇异爆破，所有这些都可以通过一个额外的纳什爆破来解决。存在一个特殊的理想列R=J_0，J_1，J_2，.使得V_i= \subset V是V的第i次Nash爆破，其中J_i|所有i的J_{i+1}。应用我们以前的文章，V_i是非奇异的当且仅当J_{i+1}的理想类整除J_i的理想类的某个幂。本文使事情变得更加现实：理想类的这种可分性意味着对于某些N \ge r+2 J_i^{N-r-2} J_{i+1} ^{r+3}=J_i^ NJ_{i+2}。但是要注意，这个恒等式又意味着J_{i+2}是J_{i+1}的某个幂的约数。因此，尽管$V_i$可能不是非奇异的，但当恒等式保持{ \it next}时，变种V_{i+1}必须是非奇异的。因此，纳什问题等价于这样一个断言，即上述恒等式对某个足够大的i和N成立。
Higher Jacobian matrix of weighted homogeneous polynomials and derivation algebras
本文证明加权齐次多项式高阶 Jacobian 矩阵极大子式所生成的理想仍为加权齐次，并据此部分回应高阶 Nash 爆破局部代数上不存在负权导子的猜想。
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