On The Negative K-Theory of Schemes in Finite Characteristic
Amalendu Krishna
本文研究了正特征域上奇异簇的负$K$ -理论，特别证明了对于维数为$d$的$k$ -簇，$K_i(X)$对$i < -d-2$为零.
We study the negative $K$-theory of singular varieties over a field of positive characteristic and in particular, prove the vanishing of $K_i(X)$ for $i < -d-2$ for a $k$-variety of dimension $d$.
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Complex projective threefolds with non-negative canonical Euler-Poincare characteristic
设$V$是一般型的复非奇异射影3-重，且有$χ(ω_V)\geq 0$（分别为. $>0$）。本文证明了m-正则映射$Φ_{|mK_V|}$在其象上是双有理的，对所有的$m\ge 14$（分别为. $\geq 8$）。已知的例子表明，下界$r_3=14$（分别为）。$=8$）是最佳的。
Vector bundles near negative curves: moduli and local Euler characteristic
研究了不可约曲线$\ell = CP^1$的二维邻域$\ell^2 = -k$上向量丛的模空间，给出了这些模作为分层空间的显式构造。给出了$Z_k$上丛的局部全纯欧拉示性数的精确界，并证明了具有给定数值不变量的丛族的存在性.我们的数值计算使用Macaulay 2算法进行，该算法可在http://www.maths.ed.ac.uk/~s0571100/Instanton/下载。
Desingularization of quasi-excellent schemes in characteristic zero
Grothendieck在EGA IV中证明了，如果局部Noether格式X上的任何有限型整概形允许去奇异化，则X是拟优的，并证明了反过来也可能成立。我们证明了这一猜想的noether格式的特征为零。即从特征为零的代数簇的奇点的分解出发，证明了Noether拟优Q格式的奇点的分解。
We compute the Euler characteristics of tautological vector bundles and their exterior powers over the Quot schemes of curves. We give closed-form expressions over punctual Quot schemes in all genera. For higher rank quotients of a trivial vector bundle, we obtain answers in genus zero. We also study the Euler characteristics of the symmetric powers of the tautological bundles, for rank zero quotients.
The Euler characteristics of generalized Kummer schemes
我们计算了与$A\times Y$相关的广义Kummer格式的欧拉示性数，其中$A$是阿贝尔簇，$Y$是光滑的拟射影簇.当$Y$是点时，我们的结果证明了Gulbrandsen猜想的公式。
We prove functorial weak factorization of projective birational morphisms of regular quasi-excellent schemes in characteristic 0 broadly based on the existing line of proof for varieties. From this general functorial statement we deduce factorization results for algebraic stacks, formal schemes, complex analytic germs, Berkovich analytic and rigid analytic spaces.
Strongly Liftable Schemes and the Kawamata-Viehweg Vanishing in Positive Characteristic
正特征域k上的光滑概形X称为强提升的，如果X和X上的所有素除子都能在W_2（k）上同时提升.本文给出了强提升格式的一些具体例子和性质。作为应用，我们证明了正特征的Kawamata-Viehweg消灭定理对任何相对于强提升曲面为双有理的法向射影曲面成立.
Hilbert schemes of a surface and Euler characteristics
利用基本代数拓扑和Ellingsrud-Stromme关于曲面的点Hilbert格式的贝蒂数的结果，计算了复曲面的n点Douady空间的欧拉示性数的生成函数.射影情形首先由L. Göttsche
A Macaulay2 package for characteristic classes and the topological Euler characteristic of complex projective schemes
Macaulay 2软件包CharacteristicClasses提供了计算复射影空间的闭子概型的拓扑欧拉示性数、Chern类的度和Segre类的度的命令。计算可以用符号和数字两种方式完成，后者使用Bertini的接口。我们提供了一些实现的背景，并展示了如何使用该包的帮助下的例子。
The tame fundamental group schemes of curves in positive characteristic
本文研究正特征代数闭域上光滑曲线的驯服基本群概形，发展余特化映射理论，证明其依赖曲线且可重构数值不变量。
We denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree $d$ and genus $g$ in $\PP^r$. In this article, we show that for low genus $g$ outside the Brill-Noether range, the Hilbert scheme $\mathcal{H}_{g+1,g,4}$ is non-empty whenever $g\ge 9$ and irreducible whose only component generically consists of linearly normal curves unless $g=9$ or $g=12$. This complements the validity of the original assertion of Severi regarding the irreducibility of $\mathcal{H}_…
Computing characteristic classes of projective schemes
本文讨论了一个算法，它计算了与一个（可能是奇异的、可约的、非可约的）射影概形相关联的几个类到射影空间的推进。例如，该算法产生的拓扑欧拉示性数的射影概形$S$的支持，给定的齐次理想的$S$。该算法已在Macaulay2中实现。
Topics in group schemes and surfaces in positive characteristic
该综述基于作者2023年在爱丁堡大学的讲座，介绍正特征下代数曲面中的无穷小群概形、群作用及Kodaira消没定理失效等现象，并辅以显式例子。
This paper summarizes the results at the present moment about singularities with respect to the Mather-Jacobian log discrepancies over algebraically closed field of arbitrary characteristic. The basic point is the Inversion of Adjunction with respect to Mather-Jacobian discrepancies holds in arbitrary characteristic. Based on this fact we will reduce many geometric properties of the singularities into the problem on jet schemes and try to avoid discussions which are distinctive for characteristic 0.
The Euler characteristic of the generalized Kummer scheme of an Abelian threefold
设$X$是阿贝尔三重数。证明了第一作者提出的用平面剖分数表示$X$的广义Kummer格式$K^nX$的欧拉示性数的公式。这将计算模堆栈$[K^nX/X_n]$的Donaldson-Thomas不变量。
Functorial desingularization of quasi-excellent schemes in characteristic zero: the non-embedded case
我们证明了任何特征为零的Noether拟优方案都有一个强去奇异化，它是关于所有正则态射的函子。我们表明，作为一个简单的正式后果，这一结果得到了强函的desingularization的许多其他空间的特征零，包括准优秀的堆栈和形式概形，复杂的和非阿基米德解析空间。此外，这些函子很容易推广到非紧的设置，利用收敛爆破超序列的正则中心。
Gromov-Witten theory of schemes in mixed characteristic
本文定义了戴德金整环上有限表示光滑射影概形的Gromov-Witten类和不变量。我们证明了它们是形变不变量，并验证了基本公理。对于戴德金整环上的光滑射影概形，我们证明了不同特征的纤维的不变量是相同的。证明了亏格为零的Gromov-Witten不变量定义了一个满足WDVV方程的势函数，并由此导出了亏格为零的Gromov-Witten不变量在任意特征下的重构定理.
We present scheme theoretic methods that apply to the study of secant varieties. This mainly concerns finite schemes and their smoothability. The theory generalises to the base fields of any characteristic, and even to non-algebraically closed fields. In particular, the smoothability of finite schemes does not depend on the embedding into a smooth variety or on base field extensions. Independent of the base field, secant varieties to high degree Veronese reembeddings behave well with respect to the intersection and they are defined by minors of catalecticants whenever a suitable smoothability …
In this paper we show that the l^n-torsion part of the cohomological Brauer groups of certain schemes associated to symmetric powers of a projective smooth curve over a separably closed field k are isomorphic, when `l is invertible in k. The schemes considered are the Symmetric powers themselves, then the corresponding Picard schemes and also certain Quot-schemes. We also obtain similar results for Prym varieties associated to certain finite covers of such curves: we prove such results only for curves defined over the field of complex numbers.
We compute tautological integrals over Quot schemes on curves and surfaces. After obtaining several explicit formulas over Quot schemes of dimension 0 quotients on curves (and finding a new symmetry), we apply the results to tautological integrals against the virtual fundamental classes of Quot schemes of dimension 0 and 1 quotients on surfaces (using also universality, torus localization, and cosection localization). The virtual Euler characteristics of Quot schemes of surfaces, a new theory parallel to the Vafa-Witten Euler characteristics of the moduli of bundles, is defined and studied. Co…
Quotients of schemes by $α_p$ or $μ_p$ actions in characteristic p>0
本文研究定义在特征p>0的域上的允许非平凡$α_p$或$μ_p$作用的方案X.特别地，商映射$X \rightarrow Y$的结构进行了研究。给出了商Y的局部性质，如奇点和局部Picard群，商映射的结构定理和商映射的附加公式。
Strongly Liftable Schemes and the Kawamata-Viehweg Vanishing in Positive Characteristic III
正特征域k上的光滑概形X称为在W_2（k）上强提升的，如果X和X上的所有素除子都能在W_2（k）上同时提升.本文首先导出了W_2（k）上的Kummer覆盖技巧，利用它可以构造出W_2（k）上的一大类光滑可提升的投射簇，并给出了强可提升格式上的Kawamata-Viehweg消灭定理的一个直接证明。其次，我们将文献[Xie 10，Xie 11]中的几乎所有结果推广到了所有的情形，即所有的情形都考虑在W（k）上，即k的Witt向量环上。
Bloch-Ogus theory for smooth and semi-stable schemes in mixed characteristic
本文研究了混合特征下光滑与半稳定概形上对偶同调理论的Bloch-Ogus理论与Gersten猜想，证明了光滑情形下的猜想及半稳定情形的特例，并由此导出优秀离散赋值环上局部环的Galois符号映射的满射性。
Universal formulas for characteristic classes on the Hilbert schemes of points on surfaces
本文可以看作是第一作者的文章“仿射平面上点的Hilbert格式上的切丛的陈类”的续篇，在这篇文章中，他通过证明用Nakajima的创造算子表示Hilbert格式的特征类的某些普遍公式的存在性，计算了仿射平面上点的Hilbert格式的总陈类.这项工作的目的（至少）是双重的。首先，我们通过定义一个生成算子的泛代数，澄清了Hilbert格式上同调公式的“泛”性概念。这有助于我们以一种非常精确的方式重新表述和扩展第一作者以前的许多结果。其次，我们能够扩展以前发现的结果，显示如何计算任何示性类的希尔伯特计划的点在仿射平面的创造运营商。特别地，我们已经包括了总Segre类和Todd类的平方根的计算。使用这些方法，我们还找到了一种计算仿射平面上点的Hilbert方案上任何重言层的任何示性类的方法。这实际上给出了仿射平面上点的希尔伯特格式的上同调空间的环结构的另一个完整描述。
Characteristic classes of Hilbert schemes of points via symmetric products
得到了任意纯维光滑拟射影簇的Hilbert点格式的Hirzebruch同调特征类的生成列（在Hilbert-Chow态射下的前推）的一个公式.这一结果是基于一个尺规作图法的动机指数推广的概念动机的权力结构，以及在一个公式的生成系列的对称产品的Hirzebruch同源特征类。我们应用相同的方法计算生成系列公式的Hirzebruch类的推进的“虚拟动机”的希尔伯特计划的三倍。作为推论，我们得到了光滑拟射影簇上的Hilbert格式的MacPherson（和Macffi）Chern类的对应结果。三倍）。对于一个投射的卡-丘三重，后者产生一个陈类版本的零维MNOP猜想。
Strongly Liftable Schemes and the Kawamata-Viehweg Vanishing in Positive Characteristic II
正特征域k上的光滑概形X称为强提升的，如果X和X上的所有素除子都能在W_2（k）上同时提升.本文首先证明了光滑复曲面簇是强提升簇。作为推论，我们得到了光滑投射环面簇的Kawamata-Viehweg消灭定理。其次，我们证明了关于强可升光滑射影曲面的双有理正规射影曲面的Kawamata-Viehweg消灭定理。最后，我们导出了W_2（k）上的循环覆盖技巧，它可以用来构造一大类可提升光滑射影簇。
Invertible top form on the Hilbert scheme of a plane in positive characteristic
本文在正特征下证明了平面Hilbert方案上存在可逆顶微分形式，并利用其局部可和性推广了Harish-Chandra定理。
We give explicit blowups of the projective plane in positive characteristic that contain smooth rational curves of arbitrarily negative self-intersection, showing that the Bounded Negativity Conjecture fails even for rational surfaces in positive characteristic.
On proper splinters in positive characteristic
该研究探讨正特征下分裂体性质的导出不变性，证明正常Gorenstein projective变种的分裂体性质与全局F-正则性为导出不变量。
McKay correspondence for linearly reductive finite group schemes in positive characteristic
该研究在任意特征下建立有限线性约化群概型的动机与上同调McKay对应，推广Artin栈动机积分理论，证明平展解消欧拉数等于群不可约表示数的结论。
We consider the problem of existence of Neron models for a family of abelian varieties over a base of dimension greater than 1. We show that when S is of equicharacteristic zero, the condition of toric additivity introduced in [Ore18] is sufficient for the existence of a Neron model, and also necessary under some extra assumptions. Furthermore, we give an equivalent formulation of toric additivity in terms of monodromy action on the l-adic Tate module.
Let $\mathbb K$ be an algebraically closed field of characteristic $0$. A curve of $(\mathbb K^*)^2$ arising from a Laurent polynomial in two variables is {\em intrinsic negative} if its tropical compactification has negative self-intersection. The aim of this note is to start a systematic study of these curves and to relate them with the problem of computing Seshadri constants of toric surfaces.
We show that, for a pseudo-proper smooth noetherian formal scheme $\mathfrak{X}$ over a positive characteristic $p$ field, its truncated De Rham complex up to the characteristic $p$ is decomposable. Moreover, if the dimension of $\mathfrak{X}$ is exactly $p$, then the full De Rham complex is decomposable. Along the way we establish the Cartier isomorphism associated to a smooth morphism of positive characteristic noetherian formal schemes.
Vanishing of negative $K$-theory in positive characteristic
我们展示了如何一个定理的Gabber的更改可以用来申请工作的Cisinski，Suslin，Voevodsky，和韦伯证明，$K_n(X)[1/p] = 0$为$n < - \dim X$其中$X$是一个准优秀的诺特计划，$p$是一个素数，是幂零的$X$，和$K_n$是$K$ -理论的巴斯-罗森-Trobaugh。这部分回答了韦伯的一个问题。
Adequate moduli spaces and geometrically reductive group schemes
本文研究特征p（及混合特征）下的充分模空间理论，通过几何可约群方案刻画商结构。
The Euler characteristic of a semialgebraic set can be considered as a generalization of the cardinality of a finite set. An advantage of semialgebraic sets is that we can define "negative sets" to be the sets with negative Euler characteristics. Applying this idea to posets, we introduce the notion of semialgebraic posets. Using "negative posets", we establish Stanley's reciprocity theorems for order polynomials at the level of Euler characteristics. We also formulate the Euler characteristic reciprocities for chromatic and flow polynomials.
Theory of weights for log convergent cohomologies II: the case of a proper SNCL scheme in characteristic $p>0$
本文在混合特征情形下，为真简单正规交叉对数概形构造了两个典范同构的滤过复形，由此导出对数收敛上同调层上的权重滤过，并建立了与作者先前等扎里斯基权重滤过复形的比较定理。
Theory of weights for log convergent cohomologies I: the case of a proper smooth scheme with an SNCD in characteristic p>0
本研究针对特征p>0下带相对简单正规交叉除子的对数概形，定义对数收敛上同调的两类基本滤过复形，证明p进纯度与二者典范同构，得到权谱序列，并建立与此前定义的Zariski侧权滤过复形的比较定理。
We show that Hilbert schemes of points on supersingular Enriques surface in characteristic 2 are simply connected, symplectic varieties but are not irreducible symplectic as the hodge number $h^{2,0} > 1$, even though a supersingular Enriques surface is an irreducible symplectic variety. These are the classes of varieties which appear only in characteristic 2 and they show that the hodge number formula for Göttsche-Soergel does not hold over characteristic 2. It also gives examples of varieties with trivial canonical class which are neither irreducible symplectic nor Calabi-Yau, thereby showin…
We present some simple examples of smooth projective varieties in positive characteristic, arising from linear algebra, which do not admit a lifting neither to characteristic zero, nor to the ring of second Witt vectors. Our first construction is the blow-up of the graph of the Frobenius morphism of a homogeneous space. The second example is a blow-up of $\mathbb{P}^3$ in a 'purely characteristic-$p$' configuration of points and lines.
We extend the recent classification of Hilbert schemes with two Borel-fixed points to arbitrary characteristic. We accomplish this by synthesizing Reeves' algorithm for generating strongly stable ideals with the basic properties of Borel-fixed ideals and our previous work classifying Hilbert schemes with unique Borel-fixed points.
The abelian fundamental group with modulus in mixed characteristic
本文在混合特征下定义了带模的交换基本群，并证明了严格半稳定概形上限制到足够大次数超曲面诱导交换基本群同构的Lefschetz型定理。
Cyclic homology, cdh-cohomology and negative K-theory
证明了一个循环同调的爆破公式，并利用它证明了无穷小$K$ -理论满足$cdh$ -下降.结合这一结果与一些计算的$cdh$ -上同调正则函数层，我们验证了一个猜想的韦伯预测消失的代数$K$ -理论的计划在度小于负的维数的计划，计划基本上的有限型在一个领域的特征为零。
Weil-étale cohomology and zeta-values of arithmetic schemes at negative integers
本文基于韦尔-étale上同调提出关于算术方案在负整数处ζ函数特例值的猜想，并证明其在特定方案上成立。
Characteristic classes of the Hilbert schemes of points on non-compact simply-connected surfaces
证明了非紧单连通曲面上点的Hilbert格式的切丛上任意乘性示性类的一个闭公式。作为推论，我们推导出了这些Hilbert格式的切丛的Chern特征标的闭公式.我们还给出了曲面上线丛的重言式丛的乘法特征类的一个闭公式。最后，我们注意到这里的结果有什么影响的希尔伯特计划的点的任意表面。
On the vanishing of negative K-groups
设k是正特征p的无限完满域，并假定奇点的强奇点的分解在k上成立。证明了：若X是d维Noether概型，其基础约化概形本质上是域k上的有限型，则对任意q < -d，负K群K_q（X）为零.这部分证实了韦伯的一个猜想。
We provide a formula to compute the big Cohen-Macaulay test ideal for triples $((R,Δ),\mathfrak{a}^{t})$ where $R$ is a mixed characteristic toric ring and $\mathfrak{a}$ is a monomial ideal. Of particular interest is that this result is consistent with the formulas for test ideals in positive characteristic and multiplier ideals in characteristic zero.
Normal hyperplane sections of normal schemes in mixed characteristic
本文研究了混合特征下正常方案的正常超平面截面，证明了在一定条件下存在无限多个正常有效Cartier除子，并应用该结果证明了除子类群限制映射的Grothendieck-Lefschetz类型结果。
Bertini theorems for singular schemes and nearby cycles in mixed characteristic
本文首先研究了域上某些奇异格式的超平面截面。证明了对数点上一类对数光滑概形的一般超平面截面的对数光滑性的Bertini定理。我们还给出了一个抽象的推广通常的贝尔蒂尼定理光滑。然后，我们应用这一结果的研究超平面截面的正则计划在一个完整的离散赋值环的特征零代数封闭的剩余域。最后，给出了Faltings的一个结果，即在混合特征为Galois上同调的离散赋值环上，可以计算光滑概形的$p$ -adic邻近环.
Unexpected subgroup schemes of PGL_{2,k} in characteristic 2
本文证明在特征 2 下 PGL_{2,k} 的无穷小子群概型不能提升到 SL_{2,k}，并完整分类了其幂幺子群概型，同时处理三角化情形。
In this article, we study the relative negative K-groups $K_{-n}(f)$ of a map $f: X \to S $ of schemes. We prove a relative version of the Weibel conjecture i.e. if $f: X \to S$ is a smooth affine map of noetherian schemes with $\dim S=d$ then $K_{-n}(f)=0$ for $n> d+1$ and the natural map $K_{-n}(f) \to K_{-n}(f \times \mathbb{A}^{r})$ is an isomorphism for all $r>0$ and $n>d.$ We also prove a vanishing result for relative negative K-groups of a subintegral map.
Zeta-values of one-dimensional arithmetic schemes at strictly negative integers
本文通过Weil-étale形式化，推导了一维算术方案在严格负整数处的ζ值公式，并将其与主要猜想联系起来。
Henselian schemes in positive characteristic
本文重新审视了 Henselian 概形理论的基础，证明了在正特征下拟凝聚层不会出现特征 0 中的病态行为，并为一般 Henselian 背景下光滑和 étale 映射的类比建立了良好性质。
Stable maps and Hurwitz schemes in mixed characteristic
本文给出了混合特征完备Hurwitz方案（和栈）的定义。我们遵循R的一个思想。Pandharipande，定义完备Hurwitz栈为亏格g的结点点曲线到亏格h的稳定点曲线的模栈上的泛曲线的d次稳定映射的子栈.这个子栈只是特征为0的Hurwitz覆盖的轨迹的闭包。我们详细描述了减少模2的完整的Hurwitz堆栈的亏格1双覆盖的射影线。结果比你想象中最狂野的梦想还要狂野。最后，我们推测一个可能的关系与工作的阿布拉莫维奇，维斯托利稳定的地图到德利涅芒福德栈。
Weil-étale cohomology and duality for arithmetic schemes in negative weights
本文研究了算术方案在负权下的 Weil-étale 上同调和对偶性，推广了 Flach 和 Morin 的构造并去除了适当的和正则的假设。
Purity results for $p$-divisible groups and abelian schemes over regular bases of mixed characteristic
让$p$是一个素数。设$(R,\ideal{m})$是混合特征$(0,p)$和分枝绝对指数$e$的正则局部环.我们提供了一般的条件时，每个阿贝尔计划在$\Spec R\setminus\{\ideal{m}\}$扩展到一个阿贝尔计划在$\Spec R$。我们表明，这样的扩展总是存在，如果$e\le p-1$，存在于大多数情况下，如果$p\le e\le 2p-3$，并不存在一般，如果$e\ge 2p-2$。$e\le p-1$的情形意味着Shimura簇在离散赋值环$O$上的积分典范模型的唯一性，该赋值环$O$具有混合特征$(0,p)$和分歧指数至多$p-1$ .这导致了大量的Néron模型在$O$上的例子。如果$p>2$和索引$p-1$，则示例是新的。
We construct characteristic classes for singular algebraic varieties in motivic Borel-Moore homology, extending the motivic Euler class of the tangent bundle defined for smooth varieties. The two classes we define refine, in the setting of motivic homotopy theory, the top degree of the class constructed by Brasselet-Schuermann-Yokura in G-theory, and the top degree of the pro-CSM class constructed by Aluffi in pro-Chow groups. We deduce an extension of the motivic Gauss-Bonnet formula to non-smooth proper varieties.
We study divided power structures on finitely generated $k$-algebras, where $k$ is a field of positive characteristic $p$. As an application we show examples of $0$-dimensional Gorenstein $k$-schemes that do not lift to a fixed noetherian local ring of non-equal characteristic. We also show that Frobenius neighbourhoods of a singular point of a general hypersurface of large dimension have no liftings to mildly ramified rings of non-equal characteristic.
We study the holomorphic Euler characteristics of tautological sheaves on Hilbert schemes of points on surfaces. In particular, we establish the rationality of K-theoretic descendent series. Our approach is to control equivariant holomorphic Euler characteristics over the Hilbert scheme of points on the affine plane. To do so, we slightly modify a Macdonald polynomial identity of Mellit.
We study the geometry and topology of Hilbert schemes of points on the orbifold surface [C^2/G], respectively the singular quotient surface C^2/G, where G is a finite subgroup of SL(2,C) of type A or D. We give a decomposition of the (equivariant) Hilbert scheme of the orbifold into affine space strata indexed by a certain combinatorial set, the set of Young walls. The generating series of Euler characteristics of Hilbert schemes of points of the singular surface of type A or D is computed in terms of an explicit formula involving a specialized character of the basic representation of the corr…
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