Edoardo Sernesi
We study the 0-th local cohomology module of the jacobian ring of a singular reduced complex projective hypersurface X, by relating it to the sheaf of logarithmic vector field along X. We investigate the analogies between the local cohomology and the well known properties of the jacobian ring of a nonsingular hypersurface. In particular we study self-duality, Hodge theoretic and Torelli type questions.
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Let C be an integral projective curve with surficial singularities. We prove that topologically trivial line bundles on the compactified Jacobian of C are in one-to-one correspondence with line bundles on C (the autoduality conjecture), and compute the cohomology of the line bundles. We also show that the natural Fourier-Mukai functor between the derived categories of quasi-coherent sheaves on the Jacobian and on the compactified Jacobian is fully faithful.
We prove that the $0$-th local cohomology of the jacobian ring of a projective hypersurface with isolated singularities has a nice interpretation it in the context of linkage theory. Roughly speaking, it represents a measure of the failure of Gherardelli's theorem for the corresponding graded modules. This leads us to a different and characteristic free proof of its self-duality, which turns out to be an easy consequence of Grothendieck's local duality theorem.
We provide an explicit presentation of the equivariant cohomology ring of the compactified Jacobian $J_{q/p}$ of the rational curve $C_{q/p}$ with planar equation $x^{q}=y^{p}$ for $(p,q)=1$. We also prove analogous results for the closely related affine Springer fiber $Sp_{q/p}$ in the affine flag variety of $SL_{p}$. We show that the perverse filtration on the cohomology of $J_{q/p}$ is multiplicative, and the associated graded ring under the perverse filtration is a degeneration of the ring of functions on a moduli space of maps $\mathbb{P}^{1}\to C_{q/p}$. We also propose several conjectur…
The Jacobian ring J(X) of a smooth hypersurface determines its isomorphism type. This has been used by Donagi and others to prove the generic global Torelli theorem for hypersurfaces in many cases. In Voisin's original proof of the global Torelli theorem for smooth cubic fourfolds the Jacobian ring does not intervene. In this paper we present a proof of the global Torelli theorem for cubic fourfolds that relies on the Jacobian ring and the (derived) global Torelli theorem for K3 surfaces. It emphasizes, once again, the relation between K3 surfaces and smooth cubic fourfolds. More generally, fo…
Let $f$ be an invertible polynomial and $G$ a group of diagonal symmetries of $f$. This note shows that the orbifold Jacobian algebra $\mathrm{Jac}(f,G)$ of $(f,G)$ defined by the authors and Elisabeth Werner in arXiv:1608.08962 is isomorphic as a $\mathbb{ZZ}/2\mathbb{ZZ}$-graded algebra to the Hochschild cohomology $\mathsf{HH}^*(\mathrm{MF}_G(f))$ of the dg-category $\mathrm{MF}_G(f)$ of $G$-equivariant matrix factorizations of $f$ by calculating the product formula of $\mathsf{HH}^*(\mathrm{MF}_G(f))$ given by Shklyarov in arXiv:1708.06030. We also discuss the relation of our previous resu…
The intrinsic cohomology ring of the universal compactified Jacobian over the moduli space of stable curves
本文研究普适紧化雅可比簇的上同调环，证明其对稳定性条件敏感，但通过迷向滤过得到与稳定性无关的内蕴上同调环，仅依赖于亏格和标记点数。
We show that the infinitesimal deformations of the Brill--Noether locus $W_d$ attached to a smooth non-hyperelliptic curve $C$ are in one-to-one correspondence with the deformations of $C$. As an application, we prove that if a Jacobian $J$ deforms together with a minimal cohomology class out the Jacobian locus, then $J$ is hyperelliptic. In particular, this provides an evidence to a conjecture of Debarre on the classification of ppavs carrying a minimal cohomology class. Finally, we also study simultaneous deformations of Fano surfaces of lines and intermediate Jacobians.
On the Cohomology of Theta Divisors of Hyperelliptic Jacobians
证明了超椭圆Jacobian中θ因子的补的第k个奇异上同调群同构于辛群Sp（2g，C）的第k个基本表示.这是我们上一篇论文中的一个结论。
Cohomology of the fine compactified universal Jacobians $\overline{J}_2$ and $\overline{J}_{2,1}$
本文计算了亏格2且带0或1个标记点的稳定曲线模空间上精细紧化万有雅可比簇的有理上同调。
Bounded cohomology property on Jacobian elliptic surfaces with simplicial Mori cones
本文针对特定雅可比椭圆曲面，证明其Mori锥为单纯的，进而推导其满足有界上同调性质，还给出满足对应条件的极小光滑射影曲面的BCP充要条件。
Minimal Cohomology Classes and Jacobians
证明了在$g$亏格的光滑曲线$C$的Jacobian $(JC,θ)$上，具有上同调类$θ^d/d!$的$JC$中的任何有效圈都是$W_{g-d}(C)$或$-W_{g-d}(C)$的平移.然后，我们用这个结果来证明，$1<d<g$，雅可比轨迹（\resp的轨迹的中间雅可比三倍）是一个不可约的组成部分的一组主要极化阿贝尔品种的尺寸$g$为$θ^d/d!$（\resp $θ^3/3!$）是一类有效的代数循环。此外，在一般三次三重环的中间雅可比行列式上，$θ^2/2!$不是有效代数环类。
Cohomology of compactified Jacobians for locally planar integral curves
本文研究局部平面整曲线紧化雅可比的上同调，探讨Ngô支持定理、perverse滤波及与希尔伯特方案的联系。
Let $J$ be the Jacobian of a smooth projective complex curve $C$ which admits non-trivial automorphisms, and let $A(J)$ be the ring of algebraic cycles on $J$ with rational coefficients modulo algebraic equivalence. We present new tautological rings in $A(J)$ which extend in a natural way the tautological ring studied by Beauville (Compos Math 140(3):683-688, 2004). We then show there exist tautological rings induced on special complementary abelian subvarieties of $J$.
We classify GV-subschemes of five-dimensional ppavs, showing that they exist only on Jacobians of curves and intermediate Jacobians of cubic threefolds, and confirming a conjecture of Pareschi--Popa in this case. The result is implied by a more general statement about subvarieties of minimal cohomology class whose sum is a theta divisor.
Smooth subvarieties of Jacobians
本文研究了在光滑拟射影复变种上存在非光滑子簇线性组合的代数整数上同调类的新例子，其中一些例子的维数为6，是最小可能的。所考虑的类是非常一般曲线Jacobian上的最小上同调类。主要工具是复 bordism。
On the tautological ring of a Jacobian modulo rational equivalence
本文研究了曲线雅可比行列式具有有理系数的Chow环。假设D是曲线的无基点g^r_d中的除数，使得典范除子K是除数D的倍数。我们发现重言式循环之间的关系。我们给出了具有p ^1的d次覆盖且其分歧点都是d阶的曲线的应用，然后给出了超椭圆曲线的应用。
In this note, we examine the Jacobian ring description of the Hodge structure of zero loci of vector bundle sections on a class of ambient varieties. We consider a set of cohomological vanishing conditions that imply such a description, and we verify these conditions for some new cases. We also observe that the method can be directly extended to log homogeneous varieties. We apply the Jacobian ring to study the null varieties of period integrals and their derivatives, generalizing a result in [9] for projective spaces. As an additional application, we prove the Hodge conjecture for very generi…
Generalized Jacobian Rings for Open Complete Intersections
本文发展了开完全交Jacobian环的理论，它是指一对$(X,Z)$，其中$X$是射影空间中的光滑完全交，$Z$是$X$中的一个简单正规交叉因子，其不可约分支是$X$上的光滑超曲面截面.我们的Jacobian环给出了开补$X-Z$的上同调的代数描述，它是Griffiths最初发明的超曲面的上同调的Poincaré剩余表示的自然推广。主要结果推广了Macaulay的对偶定理和Donagi的对称化子引理。我们的广义雅可比矩阵环区别于通常的环的一个特点是，存在对偶不完美的情况，而缺陷可以通过使用定义方程的$Z$在$X$明确控制。给出了主要结果的两个应用：一个是开完全交的无穷小Torelli问题。另一个是一个明确的界限，在完全相交的情况下，Nori的连通性。结果已被应用于研究代数周期在其他几个工程。
This note gives the complete projective classification of rational, cuspidal plane curves of degree at least 6, and having only weighted homogeneous singularities. It also sheds new light on some previous characterizations of free and nearly free curves in terms of Tjurina numbers. Finally, we suggest a stronger form of Terao's conjecture on the freeness of a line arrangement being determined by its combinatorics.
We connect two notions of tautological ring: one for the moduli space of curves (after Mumford, Faber, etc.), and the other for the Jacobian of a curve (after Beauville, Polishchuk, etc.). The motivic Lefschetz decomposition on the Jacobian side produces relations between tautological classes, leading to results about Faber's Gorenstein conjecture on the curve side. We also relate certain Gorenstein properties on both sides and verify them for small genera. Further, we raise the question whether all tautological relations are motivic, giving a possible explanation why the Gorenstein properties…
This paper studies the local geometry of compactified Jacobians constructed by Caporaso, Oda-Seshadri, Pandharipande, and Simpson. The main result is a presentation of the completed local ring of the compactified Jacobian of a nodal curve as an explicit ring of invariants described in terms of the dual graph of the curve. The authors have investigated the geometric and combinatorial properties of these rings in previous work, and consequences for compactified Jacobians are presented in this paper. Similar results are given for the local structure of the universal compactified Jacobian over the…
$I$-Maximal Variation of Hodge Structure and Jacobian Rings
本文研究光滑超曲面和完全交集家族的高阶Hodge结构变分，通过I-最大变分概念，利用Griffiths的原始上同调描述，将微分Hodge结构变分和n重Yukawa耦合解释为雅可比环的分层乘法映射，证明雅可比环的强Lefschetz性质确保I-最大变分。
We give a new method of producing relations in the tautological ring R(M_{g, 1}), using the sl_2-action on the Chow ring of the universal Jacobian. With these relations, we prove that R(M_{g, 1}) is generated by κ_i for i no greater than g/3, together with ψ. Our computation shows that Faber's conjectures for M_{g, 1} are true for g up to 19. Further, by pushing relations forward to M_g, we obtain a new proof of Faber's conjectures (for M_g) for g up to 23. For g = 24, our method recovers all the Faber-Zagier relations. We also give an algebraic proof of an identity of Morita.
Twists of intermediate Jacobian fibrations
本文通过引入解析相对雅可比层，研究OG10超凯勒拉格朗日量的截面、Tate–Shafarevich扭曲与周期，并用解析Brauer群计算扭曲群，证明三次四维流形的本原Hodge格与OG10流形第二上同调的特殊子格等距。
Endomorphiams of Polynomial Rings with Invertible Jacobians
这篇文章已被作者撤回。
Jacobian rings and the infinitesimal Torelli Theorem
本文研究Jacobian环，并识别非退化超曲面在环面中的混合Hodge组件，通过Laurent多项式计算微分的核，从而完成微分Torelli定理的证明，特别是在n≥4的情况下。
We describe several improvements to algorithms for the rigorous computation of the endomorphism ring of the Jacobian of a curve defined over a number field.
Endomorphisms of polynomial rings and Jacobians
由于第10页的一个关键论点错误，作者撤回了这篇论文。
We compare the cohomology ring of the flag variety $FL_n$ and the Chow cohomology ring of the Gelfand-Zetlin toric variety $X_{GZ}$. We show that $H^*(FL_n, \mathbb{Q})$ is the Gorenstein quotient of the subalgebra $L$ of $A^*(X_{GZ}, \mathbb{Q})$ generated by degree $1$ elements. We compute these algebras for $n=3$ to see that, in general, the subalgebra $L$ does not have Poincare duality.
We show that the image of the Abel-Jacobi map admits functorially a model over the field of definition, with the property that the Abel-Jacobi map is equivariant with respect to this model. The cohomology of this abelian variety over the base field is isomorphic as a Galois representation to the deepest part of the coniveau filtration of the cohomology of the projective variety. Moreover, we show that this model over the base field is dominated by the Albanese variety of a product of components of the Hilbert scheme of the projective variety, and thus we answer a question of Mazur. We also rec…
On the singularities of the theta divisors on Jacobians
研究了非超椭圆曲线雅可比行列式上θ因子的交上同调。
We construct an infinitesimal invariant for cycles in a family with cohomology class in the total space lying in a given level of the Leray filtration. This infinitesimal invariant detects cycles modulo algebraic equivalence in the fibers. We apply this construction to the Ikeda family, which gives optimal results for the Beauville decomposition of the 1-cycle of a very general plane curve in its Jacobian.
We characterize the relative prismatic cohomology of Bhatt and Scholze by a universal property by endowing it with the additional structure of a ``derived $δ$-ring". This involves introducing an analogue of prismatic envelopes in the setting of filtered derived commutative rings and applying this construction to the Hodge filtration on infinitesimal cohomology. In particular, we recover relative prismatic cohomology from infinitesimal cohomology via a purely algebraic process.
On quantum cohomology rings of partial flag varieties
本文的主要目的是对SL_n（C）中所有齐性空间的小量子上同调环的结构给出统一的描述。
On the Jacobian ring of a complete intersection
设f_1，. f_r是K[x_1，..]中的齐次多项式，x_n]，K为域。设F=y_1f_1+.+ y_rf_r在K[x，y]中，设I是K[x，y]的理想，由F相对于x_i和y_j的偏导数生成，F的Jacobian环是商J：=K[x，y]/I。我们通过计算某个复形的上同调群上同调群为J的复形的上同调来刻画J。
By using vector field techniques, we compute the ordinary and equivariant cohomology rings of Hilbert scheme of points in the projective plane in relation with that of a Grassmann variety.
We identify several classes of curves $C:f=0$, for which the Hilbert vector of the Jacobian module $N(f)$ can be completely determined, namely the 3-syzygy curves, the maximal Tjurina curves and the nodal curves, having only rational irreducible components. A result due to Hartshorne, on the cohomology of some rank 2 vector bundles on $\mathbb{P}^2$, is used to get a sharp lower bound for the initial degree of the Jacobian module $N(f)$, under a semistability condition.
We give a quasihomogeneity criterion for Gorenstein curves. For complete intersections, it is related to the first step of Vasconcelos' normalization algorithm. In the process, we give a simplified proof of the Kunz-Ruppert criterion.
Given a polynomial $W$ with an isolated singularity, we can consider the Jacobian ring as an invariant of the singularity. If in addition we have a group action on the polynomial ring with $W$ fixed, we are led to consider the twisted Jacobian ring which reflects the equivariant structure as well. Our main result is to show that the twisted Jacobian ring is isomorphic to an endomorphism ring of the "twisted diagonal" matrix factorization. As an application, we suggest a way to investigate Floer theory of Lagrangian submanifolds which represent homological mirror functors.
For a germ $f$ on a complex manifold $X$, we introduce a complex derived from the Liouville form acting on logarithmic differential forms, and give an exactness criterion. We use this Liouville complex to connect properties of the $D$-module generated by $f^s$ to homological data of the Jacobian ideal; specifically we show that for a large class of germs the annihilator of $f^s$ is generated by derivations. Through local cohomology, we connect the cohomology of the Milnor fiber to the Jacobian module via logarithmic differentials. In particular, we consider (not necessarily reduced) hyperplane…
Relations between tautological cycles on Jacobians
研究了曲线雅可比行列式上的重言式循环类。我们证明了一个新的结果环重言类的一般曲线，允许，除其他事项外，容易的尺寸计算，并导致一些一般的结果，这个环的结构。接下来，我们得到了消失的结果，一些生成类p_i，这给了一个改进的早期结果的Herbaut。最后，我们解除Herbaut和van der Geer-Kouvidakis的结果的Chow环（相对于其商模代数等价），我们给出了一种方法，以获得进一步明确的循环关系。作为一个成分，我们证明了一个定理如何Polishchuk的运营商D升降机的重言式子代数周（J）。
We explore the structure of the cohomology ring of the moduli space of stable 1-dimensional sheaves on $\mathbb{P}^2$ of any degree. We obtain a minimal set of tautological generators, which implies an optimal generation result for both the cohomology and the Chow ring of the moduli space. Our approach is through a geometric study of tautological relations.
Let C be a curve over a non-singular base variety S. We study algebraic cycles on the symmetric powers C^[n] and on the Jacobian J. The Chow homology of C^[*], the sum of all C^[n], is a ring using the Pontryagin product. We prove that this ring is isomorphic to CH(J)[t]<u>, the PD-polynomial algebra (variable: u) over the usual polynomial ring (variable: t) over the Chow ring CH(J). We give two such isomorphisms that over a general base are different. Further we give some precise results on how CH(J) sits embedded in CH(C^[*]) and we give an explicit geometric description of how the derivatio…
Newstead gave the generators of the cohomology ring of the moduli space of rank 2 semi-stable, torsion-free sheaves with fixed odd degree determinant over a smooth, projective curve. In this article, we generalize this result to the case when the underlying curve is irreducible, nodal. We show that these generators (of the cohomology ring in the nodal curve case) arise naturally as degeneration of Newstead's generators in the smooth curve case.
A Prym construction for the cohomology of a cubic hypersurface
Mumford定义了$P^2$上的锥丛的中间Jacobian与锥丛的判别曲线的自然定义的tale双覆盖的Prym簇之间的自然同构。Clemens和Griffiths用这个同构证明了光滑三次三重的不合理性，Beauville后来将同构推广到$P^2$上奇维四次丛的中间雅可比。我们进一步将同构推广到$P^n$中光滑三次超曲面的本原上同调。
We describe an algorithm, based on the properties of the characteristic polynomials of Frobenius, to compute $\operatorname{End}_{\overline{K}}(A)$ when $A$ is the Jacobian of a nice genus-2 curve over a number field $K$. We use this algorithm to confirm that the description of the structure of the geometric endomorphism ring of $\operatorname{Jac}(C)$ given in the LMFDB ($L$-functions and modular forms database) is correct for all the genus 2 curves $C$ currently listed in it. We also discuss the determination of the field of definition of the endomorphisms in some special cases.
The endomorphism rings of jacobians of cyclic covers of the projective line
设K是特征为0的域，$K_a$是它的代数闭包，p是奇素数.假设$f(x) \in K[x]$是一个没有重根的$n \ge 5$次多项式。让我们考虑曲线$C: y^p=f(x)$及其雅可比矩阵J（C）。已知J（C）的所有$K_a$ -自同态的环End（J（C））包含第p个分圆域（由C的明显自同构生成）中的整数环$Z[ζ_p]$。本文证明了如果K上f的伽罗瓦群是对称群$S_n$或交错群$A_n$，则有$End(J(C))=Z[ζ_p]$.
We give a short new computation of the quantum cohomology of an arbitrary smooth toric variety $X$, by showing directly that the Kodaira-Spencer map of Fukaya-Oh-Ohta-Ono defines an isomorphism onto a suitable Jacobian ring. The proof is based on the purely algebraic fact that a class of generalised Jacobian rings associated to $X$ are free as modules over the Novikov ring. In contrast to previous results of this kind, $X$ need not be compact. When $X$ is monotone the presentation we obtain is completely explicit, using only well-known computations with the standard complex structure.
Non-isogenous superelliptic jacobians
在他以前的文件（J。Math.544（2002），91- 110; Math.AG/0103203）给出了超椭圆Jacobian的一种显式构造，其自同态环是$p$第分圆域中的整数环. (Here $p$是一个奇素数。）在本文中，我们讨论当这些雅可比矩阵是相互非同构的。(The在作者的电子印刷数学中处理了超椭圆Jacobian的情况。NT/0301173。
Quantum and Floer cohomology have the same ring structure
由Floer引入的几乎Kahler流形$M$的全上同调$H^*(M)$对Floer上同调的作用给出了$H^*(M)$上的一个新的环结构.证明了具有这种新环结构的全上同调空间$H^* (M)$与量子上同调环同构。作为一个特例，我们证明了Vafa和Witten给出的复Grassmanian环的Floer上同调环的公式.
In this paper, we exploit some geometric-differential techniques to prove the strong Lefschetz property in degree $1$ for a complete intersection standard Artinian Gorenstein algebra of codimension $6$ presented by quadrics. We prove also some strong Lefschetz properties for the same kind of Artinian algebras in higher codimensions. Moreover, we analyze some loci that come naturally into the picture of "special" Artinian algebras: for them, we give some geometric descriptions and show a connection between the non emptiness of the so-called non-Lefschetz locus in degree $1$ and the "lifting" of…
Quantum Cohomology Rings of Toric Manifolds
本文计算了任意$d$维光滑投射复曲面流形${\bf P}_Σ$与扇形$Σ$的量子上同调环$H^*_φ({\bf P}, {\bf C})$. $H^*_φ({\bf P}_Σ, {\bf C})$的乘法结构取决于普通上同调群$H^2({\bf P}_Σ, {\bf C})$中元素$avarphi$的选择。量子上同调环$H^*_φ({\bf P}_Σ, {\bf C})$的许多性质被认为对Kähler流形上的量子上同调环是有效的
The singular cohomology ring of a uniform matroid: combinatorics and Lefschetz properties
本文研究均匀拟阵奇异上同调环的组合结构与Lefschetz性质，通过Koszul同调构造显式基，推导Hodge数公式并证明其满足拟射影强Lefschetz性质。
We describe the Chow rings of moduli spaces of ordered configurations of points on the projective line for arbitrary (sufficiently generic) stabilities. As an application, we exhibit such a moduli space admitting two small desingularizations with non-isomorphic cohomology rings.
The Jacobian ideal of a hyperplane arrangement
超平面排列的雅可比理想是多项式环中的理想，其生成元是定义多项式的排列的偏导数。在这篇文章中，我们证明了一个安排可以重建从它的雅可比理想。
For the split group $G_{2}$ defined over $\mathbb{Z},$ we show that the de Rham cohomology ring of $B(G_{2})_{\mathbb{F}_{2}}$ is isomorphic to the singular cohomology ring with $\mathbb{F}_{2}$-coefficients of $B(G_{2})_{\mathbb{C}}.$ For the spin groups $\textrm{Spin}(n)$ defined over $\mathbb{Z},$ we show that the de Rham cohomology ring of $B\textrm{Spin}(n)_{\mathbb{F}_{2}}$ is isomorphic to the singular cohomology ring with $\mathbb{F}_{2}$-coefficients of $B\textrm{Spin}(n)_{\mathbb{C}}$ for $n \leq 10.$ For $n=11,$ we make a full computation of the de Rham cohomology ring of $B\textrm{…
On Quantum Cohomology Rings of Fano Manifolds and a Formula of Vafa and Intriligator
我们观察到量子上同调环的一般结构定理，通常的上同调环编码信息（几乎全纯）有理曲线的非齐次版本。一个应用是严格计算格拉斯曼量子上同调。作为纯粹的代数推论，我们证明了最近引起许多数学家兴趣的Vafa和Intriligator关于从Riemann曲面（任意亏格）到G（k，n）的映射的模空间的某些紧化的交数的一个美丽公式.该公式推广到任何上同调环可表示为完全交的Fano流形。
The structure of the cohomology ring of the filt schemes
设$C$是复数域$\cnum$上的光滑射影曲线.本文研究了$\Quot(r,n)$方案的上同调环的结构，即，长度为$n$的$\nbigo_C^{\oplus r}$的商层的模方案。我们得到了$H^{\ast}(\Quot(r,n))$上的一个滤子，它的分次环具有相当简单的结构.作为推论，我们得到了环的一个小生成元。我们还得到了精确的组合描述$H^{\ast}(\Quot(r,n))$本身。为此目的，我们考虑完整的过滤器计划，我们使用“分裂原理”。一个完整滤子方案有一个简单的几何描述：一列射影空间的丛。这样就很容易确定完备滤波器格式的上同调环。在完备滤子格式的上同调环中，我们可以对“滤子格式”的上同调环进行一些计算。关键词：引文格式，上同调环，环面作用，分裂原理，对称函数。
Stability of the cohomology rings of Hilbert schemes of points on surfaces
建立了射影曲面X上n点Hilbert格式的上同调环的一些显著性质，由此可以看出这些上同调环在多大程度上依赖于X和n.
Prismatic cohomology of rigid analytic spaces over de Rham period ring
本文提出在Fontaine de Rham周期环上为具有l.c.i奇点的刚性解析空间定义prismatic上同调，扩展了相关上同调理论。
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