D. Arcara
We generalize Bertram's work on rank two vector bundles to an irreducible projective nodal curve C. We use extensions of a line bundle L by O_C and the associated `forgetful' map to study a compactification of the moduli space of semi-stable vector bundles of rank 2 and determinant L on C.
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A geometry of the moduli space of Higgs pairs on an irreducible nodal curve of arithmetic genus one
本文利用希钦映射和椭圆曲线希格斯丛模空间的平坦退化，刻画了算术亏格为一的不可约节点曲线上希格斯对模空间的几何结构。
Cohomology of Moduli Stacks of Principal $\Bbb C^*$-Bundles Over Nodal Algebraic Curves
研究结点曲线上主$\Bbb C^*$-丛模栈，用陈类确定其有理上同调代数。
We give a survey of uniformization results for principal bundles on curves. We provide a proof of uniformization for nodal curves; this result is a special case of work of Belkale and Fakhruddin for uniformization on singular curves. We use the uniformization result to give a construction of a compactification of the moduli of G bundles on a family of nodal curves. Along the way we also explain how to use equivariant compactifactions of a group to compactify the moduli stack of bundles over a fixed nodal curve.
We establish a striking connection between Abramovich's and Vistoli's twisted bundles and Gieseker vector bundles. This note grew out of an attempt to understand a recent draft of Seshadri.
Spherical vector bundles on nodal K3 surfaces
研究结点K3曲面上球面向量丛相关问题，证明K3曲面产生结点时特定陈类的稳定球面层存在性可能受阻。
Studying degenerations of moduli spaces of semistable principal bundles on smooth curves leads to the problem of constructing and studying moduli spaces on singular curves. In this note, we will see that the moduli spaces of $δ$-semistable pseudo bundles on a nodal curve constructed by the first author become, for large values of $δ$, the moduli spaces for semistable singular principal bundles. The latter are reasonable candidates for degenerations and a potential basis of further developments as on irreducible nodal curves. In particular, we find a notion of semistability for principal bundle…
Weak Brill-Noether bundles on nodal Enriques surfaces
利用节点型Enriques曲面作为Grassmannian中Reye同余的实现，研究向量丛的弱Brill-Noether问题，考虑不可约齐次丛的限制并分析其弱Brill-Noether性质和斜率稳定性。
Weil's Theorem for Logarithmic Connections on Irreducible Nodal Curves
本研究将André Weil经典定理推广至不可约射影结点曲线，证明其上不可分解向量丛或无挠凝聚层存在关于对偶化层的全纯对数联络等价于次数为0，还构造了有理结点三次曲线上的显式单参数平坦联络族。
Let $U^{'s}_L(n,d)$ be the moduli space of stable vector bundles of rank $n$ with determinant $L$ where $L$ is a fixed line bundle of degree $d$ over a nodal curve $Y$. We prove that the projective Poincare bundle on $Y \times U^{'s}_L(n,d)$ and the projective Picard bundle on $U^{'s}_L(n,d)$ are stable for suitable polarisation. For a nonsingular point $x \in Y$, we show that the restriction of the projective Poincare bundle to $x \times U^{'s}_L(n,d)$ is stable for any polarisation. We prove that for arithmetic genus $g\ge 3$ and for $g=n=2, d$ odd, the Picard group of the moduli space $U'_L…
Mehta and Seshadri have proved that the set of equivalence classes of irreducible unitary representations of the fundamental group of a punctured compact Riemann surface, can be identified with equivalence classes of stable parabolic bundles of parabolic degree zero on the compact Riemann surface. In this note, we discuss the Mehta-Seshadri correspondence over an irreducible projective curve with at most nodes as singularities.
The moduli space of Gieseker vector bundles is a compactification of moduli of vector bundles on a nodal curve. This moduli space has only normal crossing singularity and it provides a flat degeneration. We prove a Torelli type theorem for a nodal curve using the moduli space of stable Gieseker vector bundles of fixed rank (strictly greater than $1$) and fixed degree such that rank and degree are co-prime.
Let $Y_{1},\dots,Y_{l}$ be smooth irreducible projective curves and let $Y$ be its disjoint union. Given a semisimple reductive algebraic group $G$ and a faithful representation $ρ:G\hookrightarrow \textrm{SL}(V)$ we construct a projective moduli space of $(\underlineκ,δ)$-(semi)stable singular principal $G$-bundles with generalized parabolic structure of type $\underline{e}$. In case $Y$ is the normalization of a connected and reducible projective nodal curve $X$, there is a closed subscheme coarsely representing the subfunctor corresponding to descending bundles. We prove that the descent op…
Let $Y$ be an integral nodal projective curve of arithmetic genus $g\ge 2$ with $m$ nodes defined over an algebraically closed field $k$ and $x$ a nonsingular closed point of $Y$. Let $n$ and $d$ be coprime integers with $n\ge 2$. Fix a line bundle $L$ of degree $d$ on $Y$. Let $U_Y(n,d,L)$ denote the (compactified) "fixed determinant moduli space". We prove that the restriction $\mathcal{U}_{L,x}$ of the Poincare bundle to $x \times U_Y(n,d,L)$ is stable with respect to the polarisation $θ_L$ and its restriction to $x \times U'_Y(n,d,L)$, where $U'_Y(n,d,L)$ is the moduli space of vector bund…
Spinor modifications of conic bundles and derived categories of 1-nodal Fano threefolds
本文针对平坦二次曲线丛定义旋量修改构造新二次曲线丛，证明其偶克利福德代数Morita等价且导出范畴中基底导出范畴的正交补等价，并以非因子1-结点素法诺三维簇为例展示该技术，实现奇点范畴吸收。
Let $M$ be the moduli space of generalized parabolic bundles (GPBs) of rank $r$ and degree $d$ on a smooth curve $X$. Let $M_{\bar L}$ be the closure of its subset consisting of GPBs with fixed determinant ${\bar L}$. We define a moduli functor for which $M_{\bar L}$ is the coarse moduli scheme. Using the correspondence between GPBs on $X$ and torsion-free sheaves on a nodal curve $Y$ of which $X$ is a desingularization, we show that $M_{\bar L}$ can be regarded as the compactified moduli scheme of vector bundles on $Y$ with fixed determinant. We get a natural scheme structure on the closure o…
We desribe vector bundles over a class of noncommutative curves, namely, over noncommutative nodal curves of string type and of almost string type. We also prove that in other cases the classification of vector bundles over a noncommutative curve is a wild problem.
Let $X$ be a smooth projective surface and $L\in \mathrm{Pic}(X)$. We prove that if $L$ is $(2k-1)$-spanned, then the set $\tilde{V}_k(L)$ of all nodal and irreducible $D\in |L|$ with exactly $k$ nodes is irreducible. The set $\tilde{V}_k(L)$ is an open subset of a Severi variety of $|L|$, the full Severi variety parametrizing all integral $D\in |L|$ with geometric genus $g(L)-k$.
This is a review article on some applications of generalised parabolic structures to the study of torsion free sheaves and $L$-twisted Hitchin pairs on nodal curves. In particular, we survey on the relation between representations of the fundamental group of a nodal curve and the moduli spaces of generalised parabolic bundles and generalised parabolic $L$-twisted Hitchin pairs on its normalisation as well as on an analogue of the Hitchin map for generalised parabolic $L$-twisted Hitchin pairs.
A nonspecial divisor in the moduli space of cubic fourfolds via 10-nodal plane sextics
本文通过研究10节点平面六次曲线，发现立方四维流形模空间中存在一个非诺特-勒夫施茨类型的不可约除子。
Hirota Varieties and Rational Nodal Curves
本文研究了由有理节点曲线导出的Hirota簇，定义了其主分支，证明其为不可约分支等价于求解弱Schottky问题，并利用计算工具解决了直到亏格9的该问题。
Even nodal surfaces of K3 type
本文研究具有二次曲线丛结构的K3型Fano四重态，构造其与hyperKähler簇的几何联系，描述了作为Kummer四次曲面推广的节点曲面族。
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.
Under some conditions on the deformation type, which we expect to be satisfied for arbitrary irreducible symplectic varieties, we describe which big and nef line bundles on irreducible symplectic varieties have base divisors. In particular, we show that such base divisors are always irreducible and reduced. This is applied to understand the behaviour of divisorial base components of big and nef line bundles under deformations and for K3$^{[n]}$-type and Kum$^n$-type.
Let U^{'s}_{L}(n,d) be the moduli space of stable vector bundles of rank $n$ and fixed determinant L of degree d on a nodal curve Y. The moduli space of semistable vector bundles of rank n and degree $d$ will be denoted by U'_Y(n,d). We calculate the Brauer groups of U^{'s}_{L}(n,d)$. We study the question of rationality of $U^{'s}_{L}(n,d)$ and $U'_Y(n,d)$.
We give an alternative argument for the classification of real bundle pairs over smooth symmetric surfaces and extend this classification to nodal symmetric surfaces. We also classify the homotopy classes of automorphisms of real bundle pairs over symmetric surfaces. The two statements together describe the isomorphisms between real bundle pairs over symmetric surfaces up to deformation.
Globally irreducible nodes (i.e. nodes whose branches belong to the same irreducible component) have mild effects on the most common topological invariants of an algebraic curve. In other words, adding a globally irreducible node (simple nodal degeneration) to a curve should not change them a lot. In this paper we study the effect of nodal degeneration of curves on fundamental groups and show examples where simple nodal degenerations produce non-isomorphic fundamental groups and this can be detected in an algebraic way by means of Galois coverings.
This note is an attempt to generalize Bolibruch's theorem from the projective line to curves of higher genus. We show that an irreducible representation of the fundamental group of an open in a curve of higher genus has always a representation as a regular system of differential equations on a semistable bundle of degree 0. Vice-versa, we show that given such a bundle and 3 points on the curve, one can construct an irreducible representation of the curve minus the 3 points such that an associated regular system of differential equations lives on this bundle.
In this paper we study the combinatorics of quasi-trigonometric solutions of the classical Yang-Baxter equation, arising from simple vector bundles on a nodal Weierstrass cubic.
We discuss the work of Corrado Segre on nodal cubic hypersurfaces with emphasis on the cases of 6-nodal and 10-nodal cubics. In particular, we discuss the Fano surface of lines and conic bundle structures on such threefolds. We review some of the current research in algebraic geometry related to Segre's work.
We develop a new general method for computing the decomposition type of the normal bundle to a projective rational curve. This method is then used to detect and explain an example of a Hilbert scheme that parametrizes all the rational curves in $\mathbb{P}^s$ with a given decomposition type of the normal bundle and that has exactly two irreducible components. This gives a negative answer to the very old question whether such Hilbert schemes are always irreducible. We also characterize smooth non-degenerate rational curves contained in rational normal scroll surfaces in terms of the splitting t…
We consider, under suitable assumptions, the following situation: $\mathcal B$ is a component of the moduli space of polarized surfaces and $\mathcal V_{m,δ}$ is the universal Severi variety over $\mathcal B$ parametrizing pairs $(S,C)$, with $(S,H)\in \mathcal B$ and $C\in |mH|$ irreducible with exactly $δ$ nodes as singularities. The moduli map $\mathcal V\to \mathcal M_g$ of an irreducible component $\mathcal V$ of $\mathcal V_{m,δ}$ is generically of maximal rank if and only if certain cohomology vanishings hold. Assuming there are suitable semistable degenerations of the surfaces in $\mat…
A note on Severi varieties of nodal curves on Enriques surfaces
本文研究复 Enriques 曲面上 $\delta$-结点曲线的 Severi 簇，利用泛覆盖逆像曲线的可约性，确定各不可约分支在线性系中的余维为 $\delta$ 或 $\delta-1$。
It is known that the cotangent bundle $Ω_Y$ of an irreducible Hermitian symmetric space $Y$ of compact type is stable. Except for a few obvious exceptions, we show that if $X \subset Y$ is a complete intersection such that $Pic(Y) \to Pic(X)$ is surjective, then the restriction $Ω_{Y|X}$ is stable. We then address some cases where the Picard group increases by restriction.
Algebra and Geometry of Irreducible toric vector bundles of rank $n$ on $\mathbb{P}^n$
本文构造了 $\mathbb{P}^n$ 上秩 $n$ 不可约环面向量丛射影化的 Cox 环表示，并据此证明其总满足 Fujita 自由性与丰富性猜想。
Let $V$ be a $K$-vector space of dimension $n+1$. In this paper, we focus our attention into the existence of irreducible homogeneous Ulrich bundles on flag manifolds $\FF(p, q,n)$ which parameterizes all chains of linear subspaces $L_{p} \subset L_{q} \subset \PP(V)$ of dimension $p< q$, respectively. We determine all irreducible homogeneous Ulrich bundles on $\FF(0,n-1,n)$ and we prove that there are exactly $2^{n-1}$. Similarly, we prove that $\FF(0,n-2,n)$ and $\FF(1,n-1,n)$ are also the support of irreducible homogeneous Ulrich bundles. On the other hand, we prove that $\FF(0,1,n)$ do not…
For a stable irreducible curve $X$ and a torsion free sheaf $L$ on $X$ of rank one and degree $d$, D.S. Nagaraj and C.S. Seshadri ([NS]) defined a closed subset $\Cal U_X(r,L)$ in the moduli space of semistable torsion free sheaves of rank $r$ and degree $d$ on $X$. We prove that $\Cal U_X(r,L)$ is irreducible, when a smooth curve $Y$ specializes to $X$ and a line bundle $\Cal L$ on $Y$ specializes to $L$, the specialization of moduli space of semistable rank $r$ vector bundles on $Y$ with fixed determinant $\Cal L$ has underlying set $\Cal U_X(r,L)$. For rank 2 and 3, we show that there is a …
In this article, we prove the Hodge conjecture for a desingularization of the moduli space of rank 2, semi-stable, torsion-free sheaves with fixed odd degree determinant over a very general irreducible nodal curve of genus at least 2. We also compute the algebraic Poincare polynomial of the associated cohomology ring.
We investigate Koszul cohomology on irreducible nodal curves. In particular, we prove both Green and Green-Lazarsfeld conjectures for the general k-gonal nodal curve.
For a semi simple group G it is known the moduli stack of principal G-bundles over a fixed nodal curve is not complete. Finding a completion requires compactifying the group G. However it was shown in [34] that this is not sufficient to complete the moduli stack over a family of curves. In this paper I describe how to use an embedding of the loop group LG to provide a completion of the stack of G-bundles over a one dimensional family of curves degenerating to a nodal curve. The completion comes with a modular interpretation inspired by the work of Gieseker, Seshadri, Kausz and Thaddeus and Mar…
Over a family $\mathbb X$ of genus $g$ complete curves, which gives the degeneration of a smooth curve into one with nodal singularities, we build a moduli space which is the moduli space of ${\rm SL}(n, \mathbb C)$ bundles over the generic smooth curve $X_t$ in the family, and is a moduli space of bundles equipped with extra structure at the nodes for the nodal curves in the family. This moduli space is a quotient by $(\mathbb C^*)^s$ of a moduli space on the desingularisation. Taking a "maximal" degeneration of the curve into a nodal curve built from the glueing of three-pointed spheres, we …
We consider modular properties of nodal curves on general $K3$ surfaces. Let $\mathcal{K}_p$ be the moduli space of primitively polarized $K3$ surfaces $(S,L)$ of genus $p\geqslant 3$ and $\mathcal{V}_{p,m,δ}\to \mathcal{K}_p$ be the universal Severi variety of $δ$--nodal irreducible curves in $|mL|$ on $(S,L)\in \mathcal{K}_p$. We find conditions on $p, m,δ$ for the existence of an irreducible component $\mathcal{V}$ of $\mathcal{V}_{p,m,δ}$ on which the moduli map $ψ: \mathcal{V}\to \mathcal{M}_g$ (with $g= m^2 (p -1) + 1-δ$) has generically maximal rank differential. Our results, which for …
We give a presentation for the stack of rational curves with at most 1 node as the quotient by GL(3) of an open set in a 6-dimensional irreducible representation. We then use equivariant intersection theory to calculate the integral Chow ring of this stack.
On a fibre bundle version of the Caporaso-Harris formula
本文推广Caporaso-Harris公式，提出纤维丛版本，并解决P^3中平面节点三次曲线特征数计数问题。
We study the relative Hilbert scheme of a family of nodal (or smooth) curves via its (birational) cycle map, going to the relative symmetric product. We show the cycle map is the blowing up of the discriminant locus, which consists of cycles with multiple points. We derive an intersection calculus for Chern classes of tautological bundles on the relative Hilbert scheme, which has applications to enumerative geometry.
We prove that the locus of irreducible nodal curves on a given Hirzebruch surface F_k of given linear equivalency class and genus g is irreducible.
Stability conditions for line bundles on nodal curves
本文引入了节点曲线上线丛及其退化的组合稳定性条件，证明了可光滑化精细紧致雅可比簇与这些条件存在双射，并证明了精细紧致泛雅可比簇均同构于Caporaso、Pandharipande和Simpson的构造。
Let $(C,\underline{w})$ be a polarized nodal reducible curve. In this paper we consider coherent systems of type $(r,d,k)$ on $C$ with $k < r$. We prove that the moduli spaces of $(\underline{w},α)$-stable coherent systems stabilize for large $α$ and we generalize several results known for the irreducible case when we chose a good polarization. Then, we study in details the components of moduli spaces containing coherent systems arising from locally free sheaves.
Let V^{r}_{d,g, δ} be the Hilbert scheme of nodal curves in P^r of degree d and arithmetic genus g with δnodes. Under suitable numerical assumptions on d and g, for every 0 \le δ\le g we construct an irreducible component of V^{r}_{d,g, δ} having the expected number of moduli.
In this paper we focus on the problem of computing the number of moduli of the so called Severi varieties (denoted by V(|D|, δ)), which parametrize universal families of irreducible, δ-nodal curves in a complete linear system |D|, on a smooth projective surface S of general type. We determine geometrical and numerical conditions on D and numerical conditions on δensuring that such number coincides with dim(V(|D|, δ). As related facts, we also determines some sharp results concerning the geometry of some Severi varieties.
We give necessary and sufficient conditions for moduli spaces of semistable chains on a curve to be irreducible and non-empty. This gives information on the irreducible components of the nilpotent cone of GL_n-Higgs bundles and the irreducible components of moduli of systems of Hodge bundles on curves. As we do not impose coprimality restrictions, we can apply this to prove connectedness for moduli spaces of U(p,q)-Higgs bundles.
In this article, we study the deformation theory of locally free sheaves and Hitchin pairs over a nodal curve. As a special case, the infinitesimal deformation of these objects gives the tangent space of the corresponding moduli spaces, which can be used to calculate the dimension of the corresponding moduli space. We show that the deformation of locally free sheaves and Hitchin pairs over a nodal curve is equivalent to the deformation of generalized parabolic bundles and generalized parabolic Hitchin pairs over the normalization of the nodal curve respectively.
Let X be as smooth complex projective variety with Neron-Severi group isomorphic to Z, and D an irreducible divisor with normal crossing singularities. Assume r is equal to 2 or 3. We prove that if the fundamental group of X doesn't have irreducible PU(r) representations, then the fundamental group of X-D doesn't have irreducible U(r) representations. The proof uses the non-existence of certain stable parabolic bundles. We also obtain a similar result for GL(2) when D is smooth and X is a complex surface.
Varieties of Nodal surfaces, coding theory and Discriminants of cubic hypersurfaces. Part 1: Generalities and nodal K3 surfaces. Part 2: Cubic Hypersurfaces, associated discriminants. Part 3: Nodal quintics. Part 4: Nodal sextics
本文通过为射影节点曲面附加二元码，系统研究了P^3中节点曲面的Severi簇的不可约分量及其关联层级，完整分类了节点K3曲面，并揭示了Togliatti五次曲面及最大节点数六次曲面与三次超曲面判别式之间的深刻联系。
Homogeneous ACM bundles on rational homogeneous spaces
本文研究有理齐性空间上的齐性 ACM 丛与 Ulrich 丛，刻画其结构并证明相关不可约丛在扭变意义下仅有限多个，同时给出特殊情形的 ACM 数值判据。
In this short note, I point out that results of Ballico and Kool--Shende--Thomas together imply that on $K3$, Enriques, and Abelian surfaces, if $L$ is a very ample and $(2p_a(L)-2g-1)$-spanned line bundle, then the equigeneric Severi variety $V_{g}(L)$ of all curves in $|L|$ having genus $g$ is non-empty, irreducible, of the expected dimension, and its general member is a $(p_a(L)-g)$-nodal curve.
In this paper, we characterize homogeneous arithmetically Cohen-Macaulay (ACM) bundles over isotropic Grassmannians of types $B$, $C$ and $D$ in term of step matrices. We show that there are only finitely many irreducible homogeneous ACM bundles by twisting line bundles over these isotropic Grassmannians. So we classify all homogeneous ACM bundles over isotropic Grassmannians combining the results on usual Grassmannians by Costa and Mir{ó}-Roig. Moreover, if the irreducible initialized homogeneous ACM bundles correspond to some special highest weights, then they can be characterized by succinc…
We treat the problem of completing the moduli space for roots of line bundles on curves. Special attention is devoted to higher spin curves within the universal Picard scheme. Two new different constructions, both using line bundles on nodal curves as boundary points, are carried out and compared with pre-existing ones.
We classify irreducible equivariant Ulrich vector bundles on isotropic Grassmannians.
A conjecture of Mumford predicts a complete set of relations between the generators of the cohomology ring of the moduli space of rank 2 semi-stable sheaves with fixed odd degree determinant on a smooth, projective curve of genus at least 2. The conjecture was proven by Kirwan. In this article, we generalize the conjecture to the case when the underlying curve is irreducible, nodal. In fact, we show that these relations (in the nodal curve case) arise naturally as degeneration of the Mumford relations shown by Kirwan in the smooth curve case. As a byproduct, we compute the Hodge-Poincare polyn…
Let $(X,L)$ be a general primitively polarized K3 surface with $c_1(L)^2 = 2g-2$ for some integer $g \geq 2$. The Severi variety $V^{L,δ} \subset |L|$ is defined to be the locus of reduced and irreducible curves in $|L|$ with exactly $δ$ nodes and no other singularities. When $δ=g$, any curve $C \in V^{L,g}$ is a rational curve; in fact, Chen \cite{Chen02} has shown that all rational curves in $|L|$ are nodal, and the number of such rational curves is given by the Yau-Zaslow formula \cite{YZ96}. In this paper, we consider the next case where $δ= g-1$ and the Severi variety $V^{L,g-1}$ parametr…
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