On the ring of invariants of ordinary quartic curves in characteristic 2

四次曲线模栈的Chow环和上同调环

Chow and cohomology rings of moduli stacks of plane quartics

本文研究了四次曲线模栈Hacking模栈的Chow环和上同调环，通过堆叠理论加权吹胀构造了一个光滑proper Deligne-Mumford堆栈，解决了KSBA和K-模空间之间的Calabi-Yau墙交叉问题，并计算了该模栈的Poincaré多项式，证明了循环类映射在有理系数下是同构的，并确定了其Chow环的生成元和关系。

Parametrizing quartic algebras over an arbitrary base

We parametrize quartic commutative algebras over any base ring or scheme (equivalently finite, flat degree four $S$-schemes), with their cubic resolvents, by pairs of ternary quadratic forms over the base. This generalizes Bhargava's parametrization of quartic rings with their cubic resolvent rings over $\mathbb{Z}$ by pairs of integral ternary quadratic forms, as well as Casnati and Ekedahl's construction of Gorenstein quartic covers by certain rank 2 families of ternary quadratic forms. We give a geometric construction of a quartic algebra from any pair of ternary quadratic forms, and prove …

The Frobenius map, rank 2 vector bundles and Kummer's quartic surface in characteristic 2 and 3

Let X be a smooth projective curve of genus g>1 defined over an algebraically closed field k of characteristic p>0. Let M_X(r) be the moduli space of semi-stable rank r vector bundles with fixed trivial determinant. The relative Frobenius map F : X \to X_1 induces by pull-back a rational map V: M_{X_1}(r) \to M_X(r). We determine the equations of V in the following two cases (1) (g,r,p) = (2,2,2) and X non-ordinary with Hasse-Witt invariant equal to 1 (see math.AG/0005044 for the case X ordinary), and (2) (g,r,p) = (2,2,3). We also show, for any triple (g,r,p), the existence of base points of …

Invariance of quantum rings under ordinary flops: III

The paper is Part III of our ongoing project to study a case of Crepant Transformation Conjecture: K-equivalence Conjecture for ordinary flops. In this paper we prove the invariance of quantum rings for general ordinary flops, whose local models are certain non-split toric bundles over arbitrary smooth base. An essential ingredient in the proof is a quantum splitting principle, which reduces a statement in Gromov--Witten theory on non-split bundles to the case of split bundles.

三次Siegel模形式与三元四次式的不变量

Siegel modular forms of degree three and invariants of ternary quartics

本研究确定了三次Siegel模形式分次环的结构，明确其由19个模形式生成并找到7个权的齐次参数系，同时建立了三元四次式的Dixmier-Ohno不变量与这些生成元的完整对应。

Invariance of Quantum Rings under Ordinary Flops II: A quantum Leray--Hirsch theorem

This is the second of a sequence of papers proving the quantum invariance for ordinary flops over an arbitrary smooth base. In this paper, we complete the proof of the invariance of the big quantum rings under ordinary flops of splitting type. To achieve that, several new ingredients are introduced. One is a quantum Leray--Hirsch theorem for the local model (a certain toric bundle) which extends the quantum D module of Dubrovin connection on the base by a Picard--Fuchs system of the toric fibers. Nonsplit flops as well as further applications of the quantum Leray--Hirsch theorem will be discus…

The integral Chow ring of the stack of smooth non-hyperelliptic curves of genus three

We compute the integral Chow ring of the stack of smooth, non-hyperelliptic curves of genus three. We obtain this result by computing the integral Chow ring of the stack of smooth plane quartics, by means of equivariant intersection theory.

On the computation of singular plane curves and quartic surfaces

Two Magma functions are given: one computes linear systems of plane curves with non-ordinary singularities and the other computes a scheme which parametrizes given degree plane curves with given singularities. These functions provide an efficient tool to construct explicit equations of singular plane algebraic curves. By computing singular branch curves, we obtain equations of normal quartic surfaces in $\mathbb C\mathbb P^3$ having the following combinations of rational double points: $\mathsf D_5\mathsf E_7\mathsf E_7,$ $\mathsf D_7\mathsf D_6\mathsf D_6,$ $\mathsf E_6\mathsf D_8\mathsf D_5,…

具有典范移动奇点的平面四次曲线纤维化

Fibrations by plane quartic curves with a canonical moving singularity

本文分类了特征二中具有典范移动奇点的平面有理四次曲线纤维化，构造了两个万有纤维化，并详细研究了一个铅笔的几何及其极小正则模型。

Analytical invariants of quasi-ordinary hypersurface singularities associated to divisorial valuations

We study an analytically irreducible algebroid germ (X, 0) of complex singularity by considering the filtrations of its analytic algebra, and their associated graded rings, induced by the divisorial valuations associated to the irreducible components of the exceptional divisor of the normalized blow-up of the normalization (Y, 0) of (X, 0), centered at the point 0 of Y. If (X, 0) is a quasi-ordinary hypersurface singularity, we obtain that the associated graded ring is an algebra of finite type over the field of complex numbers, namely the coordinate ring of a non necessarily normal affine tor…

Invariance of Quantum Rings under Ordinary Flops I: Quantum corrections and reduction to local models

This is the first of a sequence of papers proving the quantum invariance under ordinary flops over an arbitrary smooth base. In this first part, we determine the defect of the cup product under the canonical correspondence and show that it is corrected by the small quantum product attached to the extremal ray. We then perform various reductions to reduce the problem to the local models. In Part II, we develop a quantum Leray--Hirsch theorem and use it to show that the big quantum cohomology ring is invariant under analytic continuations in the Kähler moduli space for ordinary flops of splittin…

外尔不变的\(E_8\)雅可比形式环

The ring of Weyl invariant $E_8$ Jacobi forms

研究外尔不变的\(E_8\)弱雅可比形式环，证明其与二元六次式和二元四次式联合协变量环同构且有限生成，借助已知协变量环生成元获极小生成元基。

On the cohomology rings of Grassmann varieties and Hilbert schemes

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.

Endomorphisms of ordinary superelliptic jacobians

Let $K$ be a field of prime characteristic $p$, $n>4 $ an integer, $f(x)$ an irreducible polynomial over $K$ of degree $n$, whose Galois group is either the full symmetric group $S_n$ or the alternating group $A_n$. Let $l$ be an odd prime different from $p$, $Z[ζ_l]$ the ring of integers in the $l$th cyclotomic field, $C_{f,l}:y^l=f(x)$ the corresponding superelliptic curve and $J(C_{f,l})$ its jacobian. We prove that the ring of all endomorphisms of $J(C_{f,l})$ coincides with $Z[ζ_l]$ if $J(C_{f,l})$ is an ordinary abelian variety and $(l,n)\ne (5,5)$.

Factoring Third Order Ordinary Differential Operators over Spectral Curves

We consider the classical factorization problem of a third order ordinary differential operator $L-λ$, for a spectral parameter $λ$. It is assumed that $L$ is an algebro-geometric operator, that it has a nontrivial centralizer, which can be seen as the affine ring of curve, the famous "spectral curve" $Γ$. In this work we explicitly describe the ring structure of the centralizer of $L$ and, as a consequence, we prove that $Γ$ is a space curve. In this context, the first computed example of a non-planar spectral curve arises, for an operator of this type. Based on the structure of the centraliz…

Flops, motives and invariance of quantum rings

For ordinary flops, the correspondence defined by the graph closure is shown to give equivalence of Chow motives and to preserve the Poincaré pairing. In the case of simple ordinary flops, this correspondence preserves the big quantum cohomology ring after an analytic continuation over the extended Kähler moduli space. For Mukai flops, it is shown that the birational map for the local models is deformation equivalent to isomorphisms. This implies that the birational map induces isomorphisms on the full quantum rings and all the quantum corrections attached to the extremal ray vanish.

Equivariant Cobordism of Flag Varieties and of Symmetric Varieties

We obtain an explicit presentation of the equivariant cobordism ring of a complete flag variety. An immediate corollary is a Borel presentation of the ordinary cobordism ring. Another application is an equivariant Schubert calculus in cobordism. We also describe the rational cobordism rings of wonderful symmetric varieties of minimal rank.

Computing endomorphism rings of elliptic curves under the GRH

We design a probabilistic algorithm for computing endomorphism rings of ordinary elliptic curves defined over finite fields that we prove has a subexponential runtime in the size of the base field, assuming solely the generalized Riemann hypothesis. Additionally, we improve the asymptotic complexity of previously known, heuristic, subexponential methods by describing a faster isogeny-computing routine.

常微分算子的规范型，I

Normal forms for ordinary differential operators, I

本文发展了一维广义 Schur 理论，引入微分算子规范型概念，从而获得了射影不可约曲线上上同调群消没的无挠秩 1 层的新显式参数化方法。

An equivariant basis for the cohomology of Springer fibers

Springer fibers are subvarieties of the flag variety that play an important role in combinatorics and geometric representation theory. In this paper, we analyze the equivariant cohomology of Springer fibers for $GL_n(\mathbb{C})$ using results of Kumar and Procesi that describe this equivariant cohomology as a quotient ring. We define a basis for the equivariant cohomology of a Springer fiber, generalizing a monomial basis of the ordinary cohomology defined by De Concini and Procesi and studied by Garsia and Procesi. Our construction yields a combinatorial framework with which to study the equ…

$BU(2)$ 的 $C_2$-等变普通上同调

The $C_2$-equivariant ordinary cohomology of $BU(2)$

本文计算了 $BU(2)$ 的 $C_2$-等变普通上同调（Burnside 环系数），通过扩展分次获得自然生成元，进而定义等变复二维平面丛的示性类，并结合先前工作定义等变复直线和曲面的示性数。