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 of the subset consisting of torsion-free sheaves with a fixed determinant in the moduli space of torsion-free sheaves on $Y$. The relation to Seshadri--Nagaraj conjecture is studied.
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…
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 $g(L)-k$.
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.
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 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 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…
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…
We describe new irreducible components of the moduli space of rank $2$ semistable torsion free sheaves on the three-dimensional projective space whose generic point corresponds to non-locally free sheaves whose singular locus is either 0-dimensional or consists of a line plus disjoint points. In particular, we prove that the moduli spaces of semistable sheaves with Chern classes $(c_1,c_2,c_3)=(-1,2n,0)$ and $(c_1,c_2,c_3)=(0,n,0)$ always contain at least one rational irreducible component. As an application, we prove that the number of such components grows as the second Chern class grows, an…
Brauer group and birational type of moduli spaces of torsionfree sheaves on a nodal curve
设U^{s}_{L}（n，d）是节曲线Y上秩为$n$且d次行列式为L的稳定向量丛的模空间.秩为n、度为$d$的半稳定向量丛的模空间记为U '_Y（n，d）.计算了U^{'s}_{L}（n，d）$. We study the question of rationality of $ U^{'s}_{L}（n，d）$ and $ U '_Y（n，d）$的Brauer群.
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 …
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 $(C,\underline{w})$ 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.
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
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.
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…
Let $M(2,\textbf{\underline{w}},χ)$ be the moduli space of rank $2$ torsion-free sheaves over a reducible nodal curve with each component having utmost two nodal singularities. We show that in each component of $M(2,\textbf{\underline{w}},χ)$, the closure of rank $2$ vector bundles with fixed determinant is rational.
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…
We determine the irreducible components of the space of 3x3 matrices of linear forms with vanishing determinant. We show that there are four irreducible components and we identify them concretely. In particular, under elementary row and column operations with constant coefficients, a 3x3 matrix with vanishing determinant is equivalent to one of the following four: a matrix with a zero row, a zero column, a zero 2x2 square or an antisymmetric matrix.
The surfaces considered are real, rational and have a unique smooth real $(-2)$-curve. Their canonical class $K$ is strictly negative on any other irreducible curve in the surface and $K^2>0$. For surfaces satisfying these assumptions, we suggest a certain signed count of real rational curves that belong to a given divisor class and are simply tangent to the $(-2)$-curve at each intersection point. We prove that this count provides a number which depends neither on the point constraints nor on deformation of the surface preserving the real structure and the $(-2)$-curve.
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…
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.
In this paper we answer two long-standing questions in the classification of $G$-torsors on curves for an almost simple, simply connected algebraic group $G$ over the field of complex numbers. The first question is to give an intrinsic definition of (semi)stability for a $G$-torsor on an {\em irreducible nodal curve} and the second one is the construction of a flat degeneration of the moduli space of semistable $G$-torsors when the smooth curve degenerates to an irreducible nodal curve. A generalization of the classical Bruhat-Tits group schemes to two-dimensional regular local rings and an ap…
We show that each of the irreducible components of moduli of rank 2 torsion-free sheaves with odd Euler characteristic over a reducible nodal curve is rational.
We propose a three-step program for the classification of stable rank 2 bundles on the projective space $\mathbb{P}^3$ inspired by an article by Hartshorne and Rao. While this classification program has been successfully completed for stable rank 2 bundles with even determinant and $c_2\le5$, much less is known for bundles with odd determinant. After revising the known facts about these objects, we list all possible spectra and minimal monads for stable rank 2 bundles with odd determinant and $c_2\le8$. We provide a full classification of all bundles with positive minimal monads, provide a neg…
Let $X$ be an irreducible smooth projective curve, defined over an algebraically closed field $k$, of genus at least three and $L$ a line bundle on $X$. Let ${\mathcal M}_X(r,L)$ be the moduli space of stable vector bundles on $X$ of rank $r$ and determinant $L$ with $r\geq 2$. We prove that the Brauer group ${\rm Br}(\mathcal{M}_X(r,L))$ is cyclic of order ${\rm g.c.d.}(r,{\rm degree}(L))$. We also prove that ${\rm Br}(\mathcal{M}_X(r,L))$ is generated by the class of the projective bundle obtained by restricting the universal projective bundle. These results were proved earlier in \cite{BBGN…
We study Pandharipande-Thomas's stable pair theory on $K3$ fibrations over curves with possibly nodal fibers. We describe stable pair invariants of the fiberwise irreducible curve classes in terms of Kawai-Yoshioka's formula for the Euler characteristics of moduli spaces of stable pairs on $K3$ surfaces and Noether-Lefschetz numbers of the fibration. Moreover, we investigate the relation of these invariants with the perverse (non-commutative) stable pair invariants of the $K3$ fibration. In the case that the $K3$ fibration is a projective Calabi-Yau threefold, by means of wall-crossing techniq…
