Gian Mario Besana, Sandra Di Rocco, Antonio Lanteri
Zero-schemes on smooth complex projective varieties, forcing all elements of ample and free linear systems to be reducible are studied. Relationships among the minimal length of such zero-schemes, the positivity of the line bundle associated with the linear system, and the dimension of the variety are established. A generalization to higher dimension subschemes is studied in the last section.
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The bad locus and the rude locus of an ample and base point free linear system on a smooth complex projective variety are introduced and studied. The bad locus is defined as the set of points that force divisors through them to be reducible. The rude locus is defined as the set of points such that divisors that are singular at them are forced to be reducible. The existence of a nonmempty bad locus is shown to be exclusively a two dimensional phenomenon. Polarized surfaces of small degree, or whose degree is the square of a prime, with nonempty bad loci are completely classified. Several explic…
The bad locus of a base-point free linear system L on a normal complex projective variety X is defined as the subset B(L) of X of points that are not contained in any irreducible and reduced member of L. In this paper we provide a geometric description of such locus in terms of the morphism defined by L. In particular, assume that the dimension of X is at least 2, and that L is the complete linear system associated to an ample and spanned line bundle. It is known that in this case B(L) is empty unless X is a surface. Then we prove that, when the latter occurs, B(L) is not empty if and only if …
We generalize the Gaffney-Lazarsfeld theorem on higher ramification loci of branched coverings of $\mathbb{P}^n$ to homogeneous spaces with Picard number one.
We calculate the dimension of the locus of elliptic surfaces over P^1 with a section and a given Picard number, in the corresponding moduli space.
In this paper we deal with Brill-Noether theory for higher-rank sheaves on a polarized nodal reducible curve $(C,\underline{w})$ following the ideas of [arXiv:alg-geom/9511003v1]. We study the Brill-Noether loci of $\underline{w}$-stable depth one sheaves on $C$ having rank $r$ on all irreducible components and having small slope. In analogy with what happens in the smooth case, we prove that these loci are closely related to BGN extensions. Moreover, we produce irreducible components of the expected dimension for these Brill-Noether loci.
In this paper, we prove a new expected dimension formula for certain rank two Brill-Noether loci with fixed special determinant. This answers a question asked by Osserman and also leads to a new and much simpler proof of a theorem of Osserman. Our result generalizes the well-known result by Bertram, Feinberg and independently Mukai on expected dimension of rank two Brill-Noether loci with canonical determinant and partially verifies a conjecture (in rank two) of Grzegorczyk and Newstead on coherent systems.
In this paper we construct Shimura subvarieties of dimension bigger than one of the moduli space of polarised abelian varieties of a given dimension, which are generically contained in the Pym loci of (ramified) double covers. The idea is to adapt the techniques already used to construct Shimura curves in the Prym loci to the higher dimensional case, namely to use families of Galois covers of the projective line. The case of abelian covers is treated in detail, since in this case it is possible to make explicit computations that allow to verify a sufficient condition for such a family to yield…
Higher order Petri Loci
本文研究高阶皮特里利域的结构，证明其存在 codimension k 的组件。
In this note we introduce higher order polar loci as natural generalizations of the classical polar loci, replacing the role of tangent spaces by that of higher order osculating spaces. The close connection between polar loci and dual varieties carries over to a connection between higher order polar loci and higher order dual varieties. We generalize the duality between the degrees of polar classes of a variety and those of its dual variety to varieties that are reflexive to a higher order. In particular, the degree of the top (highest codimension) polar class of order k is equal to the degree…
Osculating Geometry and Higher-Order Distance Loci
本文研究在多项式约束下优化距离函数的问题，探讨高阶距离度数的代数不变量及其在jet包和高阶极位势中的交理论意义，提出基于Bombieri-Weyl度量的闭式公式及热带框架。
Given a vector bundle $V$ over a curve $X$, we define and study a surjective rational map $\mathrm{Hilb}^d (\mathbb{P} V ) - \mathrm{Quot}^{0, d} ( V^* )$ generalising the natural map $\mathrm{Sym}^d X \to \mathrm{Quot}^{0, d} ({\mathcal O}_X)$. We then give a generalisation of the geometric Riemann--Roch theorem to vector bundles of higher rank over $X$. We use this to give a geometric description of the tangent cone to the Brill--Noether locus $B^r_{r, d}$ at a suitable bundle $E$ with $h^0 (E) = r+n$. This gives a generalisation of the Riemann--Kempf singularity theorem. As a corollary, we …
Higher characteristic classes of multisingularity loci
本文研究流形间映射的多重奇点轨迹，提出高阶 Thom 多项式的结构定理与插值定理，实现算法计算，并推导拟齐次芽像 Milnor 数的显式公式，推进 Mond 猜想。
Euler characteristics of higher rank double ramification loci in genus one
本文研究了一阶和高阶双分枝定位的欧拉特征，通过递推关系推导出闭合公式，其中一阶公式为多项式，高阶公式涉及矩阵余子式的最大公约数。
We study the base locus of the higher fundamental forms of a projective toric variety $X$ at a general point. More precisely we consider the closure $X$ of the image of a map $({\mathbb C}^*)^k\to {\mathbb P}^n$, sending $t$ to the vector of Laurent monomials with exponents $p_0,\dots,p_n\in {\mathbb Z}^k$. We prove that the $m$-th fundamental form of such an $X$ at a general point has non empty base locus if and only if the points $p_i$ lie on a suitable degree-$m$ affine hypersurface. We then restrict to the case in which the points $p_i$ are all the lattice points of a lattice polytope and …
Let $M_{\mathbb{P}^2}(v)$ be a moduli space of semistable sheaves on $\mathbb{P}^2$, and let $B^k(v) \subseteq M_{\mathbb{P}^2}(v)$ be the \textit{Brill-Noether locus} of sheaves $E$ with $h^0(\mathbb{P}^2, E) \geq k$. In this paper we develop the foundational properties of Brill-Noether loci on $\mathbb{P}^2$. Set $r = r(E)$ to be the rank and $c_1, c_2$ the Chern classes. The Brill-Noether loci have natural determinantal scheme structures and expected dimensions $dim B^k(v) = dim M_{\mathbb{P}^2}(v) - k(k - χ(E))$. When $c_1 > 0$, we show that the Brill-Noether locus $B^r(v)$ is nonempty. Wh…
Let $f \colon X \to Y$ be the blow-up of a smooth projective variety $Y$ along its codimension two smooth closed subvariety. In this paper, we show that the moduli space of stable sheaves on $X$ and $Y$ are connected by a sequence of flip-like diagrams. The result is a higher dimensional generalization of the result of Nakajima and Yoshioka, which is the case of $\dim Y=2$. As an application of our general result, we study the birational geometry of the Hilbert scheme of two points.
Martens and Mumford theorems for higher rank Brill--Noether loci
该文推广 Martens 定理，给出高秩 Brill--Noether 轨迹维数上界，证明猜想并推广 Mumford 定理，通过切空间分析，证明 $n\ge5$ 时 $B^2_{n,n(g-1)}$ 不可约且约化。
We study cohomology support loci and higher direct images of (log) pluricanonical bundles of smooth projective varieties or log canonical pairs. We prove that the 0-th cohomology support loci of log pluricanonical bundles are finite unions of torsion translates of subtori. We also construct an example of morphism from a smooth projective variety to an abelian variety such that a higher direct image of a pluricanonical bundle to the abelian variety is not a GV-sheaf.
Opers of higher types, Quot-schemes and Frobenius instability loci
本文研究了特征 p>0 光滑射影曲线上半稳定向量丛模空间的 Frobenius 不稳定轨迹，引入 q 型算子，提出其与秩为 q 的向量丛 Frobenius 直像 Quot-概形之间对应关系的猜想性推广，并给出不稳定轨迹维数的猜想公式。
Continuing our previous study of modified expected dimensions for rank-2 Brill-Noether loci with prescribed special determinants, we introduce a general framework which applies a priori for arbitrary rank, and use it to prove modified expected dimension bounds in several new cases, applying both to rank 2 and to higher rank. The main tool is the introduction of generalized alternating Grassmannians, which are the loci inside Grassmannians corresponding to subspaces which are simultaneously isotropic for a family of multilinear alternating forms on the ambient vector space. In the case of rank …
Laurent polynomial mirrors for quiver flag zero loci
本文针对高维法诺簇分类问题，将Gelfand–Cetlin退化推广到Y形箭图旗簇，提出构造法诺箭图旗零轨迹洛朗多项式镜像的方法，并验证了99个四维情形的猜想镜像。
Higher order double point formulas via SSM-Thom polynomials
本文通过SSM-Thom多项式研究高阶双点公式，推导出双点位形的修正公式并应用于奇异位形的几何分析。
Derivative and multiplicity loci of elementary symmetric polynomials in positive characteristic
本文确定了正特征下初等对称多项式所有导数与多重性轨迹的维数，修正了Orzel猜想，并分类了达到最大维数的分支，证明其一般光滑，同时给出局部环、多重性及Frobenius正规形。
Problems and related results in Algebraic Vision and Multiview Geometry
该综述梳理代数视觉与多视图几何中射影代数簇（临界轨迹）的研究进展，介绍Grassmann张量在临界轨迹理想分析与场景重构中的作用，并提出关于Grassmann张量性质及多场景临界轨迹族的长期研究方向。
In this paper, we consider higher rank Brill-Noether theory for smooth curves of genus 5, obtaining new upper bounds for non-emptiness of Brill-Noether loci and many new examples.
Avoidance Loci of Real Projective Varieties
研究避免非退化投影流形实点的实线性空间，定义并证明了避免领域的性质及其拓扑学上界。
Shifted symplectic pushforwards
本文提出沿底变换前推移位辛纤维化的方法，给出局部结构定理，并通过构建虚拟拉格朗日圈解决了Calabi-Yau 4-fold的Donaldson-Thomas理论中的形变不变性问题。
Interpolation characterization of higher Thom polynomials
本文借鉴 Maulik–Okounkov 稳定包络的插值公理，在次数有界框架内构建 SSM 类比，提出仅依靠奇异性对称性计算 SSM-Thom 多项式的新工具，并给出显式示例及几何应用。
I study a sequence of singularities in dimension 4 and above, each given by a cone of rank 1 tensors of a certain signature, which have crepant resolutions whose exceptional loci are isomorphic to cartesian powers of the projective line. In each dimension n, these resolutions naturally correspond to vertices of an (n-2)-simplex, and flops between them correspond to edges of the simplex. I show that each face of the simplex may then be associated to a certain relation between flop functors.
Brill-Noether theory on the projective plane for bundles with many sections
本文研究射影平面上半稳定层的Brill-Noether理论，用秩和斜率给出了$h^0(E)$的上界，证明了上界在Steiner丛扭曲时取得，并分类了接近上界的层，确定了相应Brill-Noether轨迹的非空性、不可约性与维数。
On the singular loci of higher secant varieties of Veronese embeddings
本文研究Veronese嵌入的高阶割线簇的奇异轨迹，通过定义m-次割线轨迹并利用射影几何与张量方法，判定其是否包含于奇异轨迹中，揭示了k值相关的三分性，并给出第4阶割线簇的具体分析。
Fixed-locus relations and purely non-symplectic automorphisms of order 6 on K3 surfaces
本文研究K3曲面上6阶纯非辛自同构的固定轨迹关系，通过比较弦性欧拉特征与霍奇数推导关系，并利用局部条件将候选数从817减少到204，且给出20种固定轨迹的显式模型。
The higher direct image complex of a coherent sheaf (or finite complex of coherent sheaves) under a projective morphism is a fundamental construction that can be defined via a Cech complex or an injective resolution, both inherently infinite constructions. Using exterior algebras and relative versions of theorems of Beilinson and Bernstein-Gel'fand-Gel'fand, we give an alternate description in finite terms. Using this description we can characterize the generic complex, over the variety of finite free complexes of a given shape, as the direct image of an easily-described vector bundle. We can …
Given a smooth projective complex curve $X$ with an involution $σ$, we study the Hitchin systems for the locus of anti-invariant (resp. invariant) stable vector bundles over $X$ under $σ$. Using these integrable systems and the theory of the nilpotent cone, we study the irreducibility of these loci. The anti-invariant locus can be thought of as a generalisation of Prym varieties to higher rank.
We extend a result by Fulton-Harris-Lazarsfeld in Brill-Noehter theory of line bundles and, as well, a result by Aprod-Sernesi in theory of Secant Loci, to the Brill-Noehter locus of stable bundles inside the moduli space of higher rank stable vector bundles on a smooth projective algebraic curve. We give some consequences of this extended result.
Lie weights for CM curves and unipotent Selmer dimensions
该研究针对几何CM雅可比的亏格≥2的Q上光滑射影曲线，证明了Bloch-Kato Selmer概形的维数不等式，回答了Kim猜想的特殊情形，核心方法基于Iwasawa理论与特征重数界估。
Bad Reduction of Genus Two Curves with complex multiplication
本文研究具有双二次CM域复乘的亏格二曲线，通过正交Shimura簇上的CM环显式描述其雅可比簇，推导稳定坏约化素数的上界并证明其存在性。
Product formulas for the Higher Bessel functions
本文研究阶乘倒数 N 次幂生成函数的乘积公式，揭示其与代数超曲面周期、Buchstaber-Rees 多项式及 N 值群律的联系，并通过广义 Frobenius 方法给出乘法核展开，进而探讨 N-Bessel 核与回文单峰多项式层级的关系。
We consider the question: ``How bad can the deformation space of an object be?'' The answer seems to be: ``Unless there is some a priori reason otherwise, the deformation space may be as bad as possible.'' We show this for a number of important moduli spaces. More precisely, every singularity of finite type over Z (up to smooth parameters) appears on: the Hilbert scheme of curves in projective space; and the moduli spaces of smooth projective general-type surfaces (or higher-dimensional varieties), plane curves with nodes and cusps, stable sheaves, isolated threefold singularities, and more. T…
We study the K-theory of actions of diagonalizable group schemes on noetherian regular separated algebraic spaces: our main result shows how to reconstruct the K-theory ring of such an action from the K-theory rings of the loci where the stabilizers have constant dimension. We apply this to the calculation of the equivariant K-theory of toric varieties, and give conditions under which the Merkurjev spectral sequence degenerates, so that the equivariant K-theory ring determines the ordinary K-theory ring. We also prove a very refined localization theorem for actions of this type.
Higher resonance schemes and Koszul modules of simplicial complexes
本文研究有限单纯复形外Stanley-Reisner代数的共振概型与高阶Koszul模，证明共振概型既约，计算Hilbert级数并给出正则性与投射维数界，推广直角Artin群Chen秩公式。
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