In this paper we study the minimal surfaces of general type with $p_g=q=1$ and $K^2=4$ whose Albanese general fibre has genus 2, classifying those such that the direct image (under the Albanese morphism) of the bicanonical sheaf is sum of line bundles. We find 8 unirational families, all of dimension strictly bigger than the expected one. These families are pairwise disjoint irreducible components of the moduli space of minimal surfaces of general type.
In this article, we classify the irreducible components of moduli stacks of torsion free sheaves of rank 2 on K3 surfaces of Picard number 1. For ruled surfaces, the components of moduli stacks of torsion free sheaves were classified by C.Walter. Moreover, by virtue of our result, we classify the irreducible components of Brill-Noether loci of Hilbert schemes of points on K3 surfaces.
We show that the moduli space of polarized irreducible symplectic manifolds of $K3^{[n]}$-type, of fixed polarization type, is not always connected. This can be derived as a consequence of Eyal Markman's characterization of polarized parallel-transport operators of $K3^{[n]}$-type.
We show that the moduli spaces of sheaves on a projective K3 surface are irreducible symplectic varieties, and that the same holds for the fibers of the Albanese map of moduli spaces of sheaves on an Abelian surface.
In this paper we investigate the asymptotic growth of the number of irreducible and connected components of the moduli space of surfaces of general type corresponding to certain families of surfaces isogenous to a higher product with group $(\mathbb Z/2\mathbb Z)^k$. We obtain a significantly higher growth than the one in our previous paper [LP14, arXiv:1402.6438].
Moduli spaces of polarised (with fixed polarisation type) irreducible holomorphic symplectic manifolds deformation equivalent to Hilbert schemes of points on $K3$ surfaces are not connected in general and A. Apostolov counted the number of their components. We extend Apostolov's results to the case of manifolds deformation equivalent to generalised Kummer varieties.
Number of moduli of irreducible families of plane curves with nodes and cusps
考虑具有d个结点和k个尖点的n次不可约平面曲线族S。设W是S的不可约分支。研究了亏格为g=（n-1）（n-2）/2-d-k的曲线的模空间与W的自然有理映射.我们定义“W的模数”为W关于这个映射的像的维数。若W的期望维数为3 n +g-1-k，则W的模数至多等于min（3g-3，3g-3+ρ-k），dove ρ是亏格为g的平滑曲线上n次2维线性级数的Brill-Neother数.如果等式成立，我们说W具有期望的模数。本文构造了一类以结点和尖点为奇点的不可约平面曲线族的例子，这些曲线具有期望的模数和非正的Brill-Noether数。
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…
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.
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.
We study the irreducible components of the moduli space of instanton sheaves on $\mathbb{P}^3$, that is rank 2 torsion free sheaves $E$ with $c_1(E)=c_3(E)=0$ satisfying $h^1(E(-2))=h^2(E(-2))=0$. In particular, we classify all instanton sheaves with $c_2(E)\le4$, describing all the irreducible components of their moduli space. A key ingredient for our argument is the study of the moduli space ${\mathcal T}(d)$ of stable sheaves on $\mathbb{P}^3$ with Hilbert polynomial $P(t)=d\cdot t$, which contains, as an open subset, the moduli space of rank 0 instanton sheaves of multiplicity $d$; we desc…
Moduli spaces of stably irreducible sheaves on Kodaira surfaces belong to the short list of examples of smooth and compact holomorphic symplectic manifolds, and it is not yet known how they fit into the classification of holomorphic symplectic manifolds by deformation type. This paper studies a natural Lagrangian fibration on these moduli spaces to determine that they are not Kähler or simply connected, ruling out most of the known deformation types of holomorphic symplectic manifolds.
On moduli spaces of vector bundles on $K3^{[n]}$-type IHS manifolds
SISSA（的里雅斯特国际高等研究学校）；Dipartimento di Matematica (DIMA)（都灵大学数学系）；Facultat de Matemàtiques i Informàtica（巴塞罗那大学数学与计算机学院）；Università degli Studi di Roma Tor Vergata（罗马二大）；Université Paris-Saclay（巴黎萨克雷大学）
We give a combinatorial description of all affine spherical varieties with prescribed weight monoid $Γ$. As an application, we obtain a characterization of the irreducible components of Alexeev and Brion's moduli scheme $\mathrm M_Γ$ for such varieties. Moreover, we find several sufficient conditions for $\mathrm M_Γ$ to be irreducible and exhibit several examples where $\mathrm M_Γ$ is reducible. Finally, we provide examples of non-reduced $\mathrm M_Γ$.
We prove that, for any $g \geq 2$, the étale double cover $ρ_g:\mathcal{E}_{g} \to \widehat{\mathcal{E}}_{g}$ from the moduli space $\mathcal{E}_{g}$ of complex polarized genus $g$ Enriques surfaces to the moduli space $\widehat{\mathcal{E}}_{g}$ of numerically polarized genus $g$ Enriques surfaces is disconnected precisely over irreducible components of $\widehat{\mathcal{E}}_{g}$ parametrizing $2$-divisible classes, answering a question of Gritsenko and Hulek. We characterize all irreducible components of $\mathcal{E}_{g}$ in terms of a new invariant of line bundles on Enriques surfaces that…
This paper is concerned with non-symplectic involutions of irreducible symplectic manifolds of $K3^{[n]}$-type. We will give a criterion for deformation equivalence and use this to give a lattice-theoretic description of all deformation types. While moduli spaces of $K3^{[n]}$-type manifolds with non-symplectic involutions are not necessarily Hausdorff, we will construct quasi-projective moduli spaces for a certain well-behaved class of such pairs.
We revisit the work of Lehn-Lehn-Sorger-van Straten on twisted cubic curves in a cubic fourfold not containing a plane in terms of moduli spaces. We show that the blow-up $Z'$ along the cubic of the irreducible holomorphic symplectic eightfold $Z$, described by the four authors, is isomorphic to an irreducible component of a moduli space of Gieseker stable torsion sheaves or rank three torsion free sheaves. For a very general such cubic fourfold, we show that $Z$ is isomorphic to a connected component of a moduli space of tilt-stable objects in the derived category and to a moduli space of Bri…
We show the existence of semiorthogonal decompositions (SOD) of Pandharipande-Thomas (PT) stable pair moduli spaces on Calabi-Yau 3-folds with irreducible curve classes, assuming relevant moduli spaces are non-singular. The above result is motivated by categorifications of wall-crossing formula of PT invariants in the derived category, and also a d-critical analogue of Bondal-Orlov, Kawamata's D/K equivalence conjecture. We also give SOD of stable pair moduli spaces on K3 surfaces, which categorifies Kawai-Yoshioka's formula proving Katz-Klemm-Vafa formula for PT invariants on K3 surfaces with…
As a continuation of the work of Freiermuth and Trautmann, we study the geometry of the moduli space of stable sheaves on $\mathbb{P}^3$ with Hilbert polynomial $4m+1$. The moduli space has three irreducible components whose generic elements are, respectively, sheaves supported on rational quartic curves, on elliptic quartic curves, or on planar quartic curves. The main idea of the proof is to relate the moduli space with the Hilbert scheme of curves by wall crossing. We present all stable sheaves contained in the intersections of the three irreducible components. We also classify stable sheav…
The moduli space of K3 surfaces $X$ with a purely non-symplectic automorphism $σ$ of order $n\geq 2$ is one dimensional exactly when $φ(n)=8$ or $10$. In this paper we classify and give explicit equations for the very general members $(X,σ)$ of the irreducible components of maximal dimension of such moduli spaces. In particular we show that there is a unique one-dimensional component for $n=20,22, 24$, three irreducible components for $n=15$ and two components in the remaining cases.
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…
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.
This work is a PhD thesis. First we provide some general context on wonderful varieties and moduli spaces of rational curves. Working over complex numbers we prove that the moduli space of rational curves with no marked points on the wonderful compactification of a symmetric space is not irreducible in general. Lastly we show that in the case of wonderful group compactifications the set of rational curves with no marked points and irreducible source is irreducible.
In this note we derive from deep results due to Clozel-Ullmo the density of Noether-Lefschetz loci inside the moduli space of marked (polarized) irreducible holomorphic symplectic (IHS) varieties. In particular we obtain the density of Hilbert schemes of points on projective $K3$ surfaces and of projective generalized Kummer varieties in their moduli spaces. We present applications to the existence of rational curves on projective deformations of such varieties, to the study of relevant cones of divisors, and a refinement of Hassett's result on cubic fourfolds whose Fano variety of lines is is…
We investigate the moduli space of sheaves supported on space curves of degree 4 and having Euler characteristic 1. We give an elementary proof of the fact that this moduli space consists of three irreducible components.
We show that there is a good notion of irreducible sympelectic varieties of $\mathrm{K3}^{[n]}$-type over an arbitrary field of characteristic zero or $p > n + 1$. Then we construct mixed characteristic moduli spaces for these varieties. Our main result is a generalization of Ogus' crystalline Torelli theorem for supersingular K3 surfaces. For applications, we answer a slight variant of a question asked by F. Charles on moduli spaces of sheaves on K3 surfaces and give a crystalline Torelli theorem for supersingular cubic fourfolds.
We study the moduli space of minimal surfaces of general type with $K_S^2 = 1$ and $p_g = 2$ and show that it is irreducible, has dimension $28$ and admits a compactification which is unirational.
Let $C$ be an elliptic curve, $w\in C$, and let $S\subset C$ be a finite subset of cardinality at least $3$. We prove a Torelli type theorem for the moduli space of rank two parabolic vector bundles with determinant line bundle $\mathcal O_C(w)$ over $(C,S)$ which are semistable with respect to a weight vector $\big(\frac{1}{2}, \dots, \frac{1}{2}\big)$.
We show that a moduli space of slope-stable sheaves over a K3 surface is an irreducible hyperkähler manifold if and only if its second Betti number is the sum of its Hodge numbers $h^{2,0}$, $h^{1,1}$ and $h^{0,2}$.
We construct quasi-projective moduli spaces of $K$-general lattice polarized irreducible holomorphic symplectic manifolds. Moreover, we study their Baily--Borel compactification and investigate a relation between one-dimensional boundary components and equivalence classes of rational Lagrangian fibrations defined on mirror manifolds.
In this note we prove that the moduli space of torsion-free modules of rank one over an Azumaya algebra on a K3-surface is an irreducible symplectic variety deformation equivalent to a Hilbert scheme of points on the K3-surface.
We construct a new infinite series of irreducible components of the Gieseker-Maruyama moduli scheme $\mathcal{M}(k), ~ k \geq 3$ of coherent semistable rank 2 sheaves with Chern classes $c_1=0,~ c_2=k,~ c_3=0$ on $\mathbb{P}^3$ whose general points are sheaves with singularities of mixed dimension. These sheaves are constructed by elementary transformations of stable and properly $μ$-semistable reflexive sheaves along disjoint union of collections of points and smooth irreducible curves which are rational or complete intersection curves. As a special member of this series we obtain a new compo…
The aim of this paper is to determine a bound of the dimension of an irreducible component of the Hilbert scheme of the moduli space of torsion-free sheaves on surfaces. Let $X$ be a non-singular irreducible complex surface and let $E$ be a vector bundle of rank $n$ on $X$. We use the $m$-elementary transformation of $E$ at a point $x \in X$ to show that there exists an embedding from the Grassmannian variety $\mathbb{G}(E_x,m)$ into the moduli space of torsion-free sheaves $\mathfrak{M}_{X,H}(n;c_1,c_2+m)$ which induces an injective morphism from $X \times M_{X,H}(n;c_1,c_2)$ to $Hilb_{\, \ma…
Let $X$ be a smooth irreducible projective curve with an involution $σ$. A vector bundle $E$ over $X$ is called anti-invariant if there exists an isomorphism $σ^*E\rightarrow E^*$. In this paper, we give a construction of the moduli spaces of anti-invariant vector bundles over $X$.
We prove the existence of a universal family over every component of the moduli space of marked irreducible holomorphic symplectic manifolds. The analogous result follows for the Teichmuller spaces.
