Shinnosuke Okawa, Hokuto Uehara
We investigate exceptional sheaves on the Hirzebruch surface $\mathbb{F}_2$, as the first attempt toward the classification of exceptional objects on weak del Pezzo surfaces.
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We axiomatize the algebraic properties of toroidal compactifications of (mixed) Shimura varieties and their automorphic vector bundles. A notion of generalized automorphic sheaf is proposed which includes sheaves of (meromorphic) sections of automorphic vector bundles with prescribed vanishing and pole orders along strata in the compactification, and their quotients. These include, for instance, sheaves of Jacobi forms and weakly holomorphic modular forms. Using this machinery we give a short and purely algebraic proof of the proportionality theorem of Hirzebruch and Mumford.
We define monads for framed torsion-free sheaves on Hirzebruch surfaces and use them to construct moduli spaces for these objects. These moduli spaces are smooth algebraic varieties, and we show that they are fine by constructing a universal monad.
We provide a stacky fan description of the total space of certain split vector bundles, as well as their projectivization, over toric Deligne-Mumford stacks. We then specialize to the case of Hirzebruch orbifold $\mathcal{H}_{r}^{ab}$ obtained by projectivizing $\mathcal{O} \oplus \mathcal{O}(r)$ over the weighted projective line $\mathbb{P}(a,b)$. Next, we give a combinatorial description of toric sheaves on $\mathcal{H}_{r}^{ab}$ and investigate their basic properties. With fixed choice of polarization and a generating sheaf, we describe the fixed point locus of the moduli scheme of $μ$-stab…
Let $X$ be a Hirzebruch surface, and let $H$ be any ample divisor. In this paper, we algorithmically determine when the moduli space of semistable sheaves $M_{X,H}(r,c_1,c_2)$ is nonempty. Our algorithm relies on certain stacks of prioritary sheaves. We first solve the existence problem for these stacks and then algorithmically determine the Harder-Narasimhan filtration of the general sheaf in the stack. In particular, semistable sheaves exist if and only if the Harder-Narasimhan filtration has length one. We then study sharp Bogomolov inequalities $Δ\geq δ_H(c_1/r)$ for the discriminants of s…
Algebraic Exceptional Set of a Three-Component Curve on Hirzebruch Surfaces
研究Hirzebruch曲面上具有正规交叉的三分量曲线B的代数例外集，证明当K+B大且B无纤维或负自交有理曲线时，该集合有限并给出有效界，且等于超双切曲线集。
We perform a study of the moduli space of framed torsion-free sheaves on Hirzebruch surfaces by using localization techniques. We discuss some general properties of this moduli space by studying it in the framework of Huybrechts-Lehn theory of framed modules. We classify the fixed points under a toric action on the moduli space, and use this to compute the Poincare polynomial of the latter. This will imply that the moduli spaces we are considering are irreducible. We also consider fractional first Chern classes, which means that we are extending our computation to a stacky deformation of a Hir…
Orbifold Hirzebruch-Riemann-Roch for quotient Deligne-Mumford stacks and equivariant moduli theory on $ K3 $ surfaces
本文通过轨形Mukai配对导出新的轨形HRR公式，用于计算K3曲面上有限群作用下稳定层等变模空间的维数，并证明这些模空间是形变等价于点Hilbert概型的不可约辛流形。
In this paper we study the connection between rigid sheaves and separable-exceptional objects on Fano varieties over arbitrary fields. We give criteria for a rigid vector bundle on a Fano variety to be the direct sum of separable-exceptional bundles and for a separable-exceptional vector bundle to be part of a full separable-exceptional collection.
Let $X$ be a smooth projective surface with a full strong exceptional sequence $\mathfrak{E}$. Under certain conditions, we describe the moduli spaces of framed sheaves on a line in $X$ via linear data, i.e. by realizing them as principal bundles over a stack of representations of the bound quiver associated to $\mathfrak{E}$.
Curves on Hirzebruch Surfaces and Semistability
研究Hirzebruch曲面上一般带点曲线的插值问题，证明除唯一例外外均不能通过预期数量的点，并用已知的$\mathbb{P}^1$几何Tevelev次数确定例外情形的插值映射数。
We compute Betti numbers of the moduli spaces of arbitrary rank stable sheaves on ruled surfaces. Our result generalizes the formula of Goettsche for rank one sheaves and the formula of Yoshioka for rank two sheaves. It also confirms the conjecture of Manschot for arbitrary rank sheaves on the Hirzebruch surfaces.
We investigate torsion exceptional sheaves on a weak del Pezzo surface of degree greater than two whose anticanonical model has at most $A_n$-singularities. We show that every torsion exceptional sheaf can be obtained from a line bundle on a $(-1)$-curve by acting spherical twist functors successively.
BPS invariants are computed, capturing topological invariants of moduli spaces of semi-stable sheaves on rational surfaces. For a suitable stability condition, it is proposed that the generating function of BPS invariants of a Hirzebruch surface takes the form of a product formula. BPS invariants for other stability conditions and other rational surfaces are obtained using Harder-Narasimhan filtrations and the blow-up formula. Explicit expressions are given for rank <4 sheaves on a Hirzebruch surface or the projective plane. The applied techniques can be applied iteratively to compute invarian…
We consider the singuralities of 2-dimensional moduli spaces of semi-stable sheaves on K3 surfaces. We show that the moduli space is normal, in particular the singuralities are rational double points. We also describe the exceptional locus on the resolution in terms of exceptional sheaves.
A real 2-elementary K3 surfaces of type ((3,1,1),- id) yields a real anti-bicanonical curve s \cup A^\prime_1 (disjoint union) on the 4-th real Hirzebruch surface F_4 where s is the exceptional section of F_4 and the real curve A^\prime_1 has one real double point. We give a criterion (see Proposition 2.4) which determines whether the real double point is degenerate or not. One direction of the assertion of this proposition has already been proved in Lemma 4.6 of the preceding paper [9] (2015) by the author. In this paper we prove the inverse direction.
We perform a study of the moduli space of framed torsion free sheaves on Hirzebruch surfaces by using localization techniques. After discussing general properties of this moduli space, we classify its fixed points under the appropriate toric action and compute its Poincare' polynomial. From the physical viewpoint, our results provide the partition function of N=4 Vafa-Witten theory on Hirzebruch surfaces, which is relevant in black hole entropy counting problems according to a conjecture due to Ooguri, Strominger and Vafa.
We show that the derived category of coherent sheaves on the quotient stack of the affine plane by a finite small subgroup of the general linear group is obtained from the derived category of coherent sheaves on the minimal resolution by adding a semiorthogonal summand with a full exceptional collection. The proof is based on an explicit construction in the abelian case, together with the analysis of the behavior of the derived categories of coherent sheaves under root constructions.
In this article we consider exceptional sequences of invertible sheaves on smooth complete rational surfaces. We show that to every such sequence one can associate a smooth complete toric surface in a canonical way. We use this structural result to prove various theorems on exceptional and strongly exceptional sequences of invertible sheaves on rational surfaces. We construct full strongly exceptional sequences for a large class of rational surfaces. For the case of toric surfaces we give a complete classification of full strongly exceptional sequences of invertible sheaves.
In this paper we prove first a general theorem on semiorthogonal decompositions in derived categories of coherent sheaves for flat families over a smooth base. Based on the results of math.AG/0510670, we then show that the derived categories of coherent sheaves on flag varieties of classical type are generated by complete exceptional collections. Finally, we find complete exceptional collections in the derived categories of some homogeneous spaces of the symplectic groups of small rank.
In the first part of this paper we provide a survey of some fundamental results about moduli spaces of framed sheaves on smooth projective surfaces. In particular, we outline a result by Bruzzo and Markushevich, and discuss a few significant examples. The moduli spaces of framed sheaves on $\mathbb{P}^2$, on multiple blowup of $\mathbb{P}^2$ are described in terms of ADHM data and, when this characterization is available, as quiver varieties. The second part is devoted to a detailed study of framed sheaves on the Hirzebruch surface $Σ_n$ in the case when the invariant expressing the necessary …
We construct a full exceptional collection of vector bundles in the derived category of coherent sheaves on the Grassmannian of isotropic two-dimensional subspaces in a symplectic vector space of dimension $2n$ and in the derived category of coherent sheaves on the Grassmannian of isotropic two-dimensional subspaces in an orthogonal vector space of dimension $2n + 1$ for all $n$.
Seshadri-type constants and Newton-Okounkov bodies for non-positive at infinity valuations of Hirzebruch surfaces
本文研究Hirzebruch曲面上由无穷远处非正除子赋值定义的旗，计算相应Seshadri型常数并显式确定Newton-Okounkov体的顶点，证明其为四边形或三角形。
The cone of curves and the Cox ring of rational surfaces over Hirzebruch surfaces
本文研究了Hirzebruch曲面上有理曲面的曲线锥与Cox环，证明了当参数δ达到特定阈值时，曲线锥为有限多面体且极小生成，Cox环有限生成，且阈值仅依赖于例外因子的组合图结构。
Let X be an algebraic variety with an action of an algebraic group G. Suppose X has a full exceptional collection of sheaves, and these sheaves are invariant under the action of the group. We construct a semiorthogonal decomposition of bounded derived category of G-equivariant coherent sheaves on X into components, equivalent to derived categories of twisted representations of the group. If the group is finite or reductive over the algebraically closed field of zero characteristic, this gives a full exceptional collection in the derived equivariant category. We apply our results to particular …
In this article we survey recent results of joint work with Lutz Hille on exceptional sequences of invertible sheaves on rational surfaces and give examples.
The paper consists of three parts. In the first of them different kinds stability are discussed. In particular, the stability concept with respect to nef divisor is introduced. A structure of rigid and superrigid vector bundles on smooth projective surfaces with nef anticanonical class is studied in the second part. We prove that any superrigid bundle has a unique exceptional filtration. In the last part we give constructive description of exceptional bundles on these surfaces.
In math.AG/0005152 a certain $t$-structure on the derived category of equivariant coherent sheaves on the nil-cone of a simple complex algebraic group was introduced (the so-called perverse $t$-structure corresponding to the middle perversity). In the present note we show that the same $t$-structure can be obtained from a natural quasi-exceptional set generating this derived category. As a consequence we obtain a bijection between the sets of dominant weights and pairs consisting of a nilpotent orbit, and an irreducible representation of the centralizer of this element, conjectured by Lusztig …
A moduli space of sheaves satisfies weak Brill-Noether if the general sheaf in the moduli space has no cohomology. Goettsche and Hirschowitz prove that on the projective plane every moduli space of Gieseker semistable sheaves of rank at least two and Euler characteristic zero satisfies weak Brill-Noether. In this paper, we give sufficient conditions for weak Brill-Noether to hold on rational surfaces. We completely characterize Chern characters on Hirzebruch surfaces for which weak Brill-Noether holds. We also prove that on a del Pezzo surface of degree at least 4 weak Brill-Noether holds if t…
In the 1980s Drézet and Le Potier realized moduli spaces of Gieseker-semistable sheaves on $\mathbb{P}^2$ as what are now called quiver moduli spaces. We discuss how this construction can be understood using t-structures and exceptional collections on derived categories, and how it can be extended to a similar result on $\mathbb{P}^1\times\mathbb{P}^1$.
Let M(v) be the moduli of stable sheaves on K3 surfaces X of Mukai vector v. If v is primitive, than it is expected that M(v) is deformation equivalent to some Hilbert scheme and weight two hogde structure can be described by H^*(X,Z). These are known by Mukai, O'Grady and Huybrechts if rank is 1 or 2, or the first Chern class is primitive. Under some conditions on the dimension of M(v), we shall show that these assertion are true. For the proof, we shall use Huybrechts's results on symplectic manifolds.
We present a method based on mutations of helices which leads to the construction (in the large volume limit) of exceptional coherent sheaves associated with the $(\sum_al_a=0)$ orbits in Gepner models. This is explicitly verified for a few examples including some cases where the ambient weighted projective space has singularities not inherited by the Calabi-Yau hypersurface. The method is based on two conjectures which lead to the analog,in the general case, of the Beilinson quiver for $\BP^n$. We discuss how one recovers the McKay quiver using the gauged linear sigma model (GLSM) near the or…
Number of full exceptional collections modulo spherical twists for elliptic orbifolds
本文通过递归公式计算了零轨形欧拉示性数轨形射影直线上凝聚层范畴中模球面扭转作用的完整例外收集数目，推广了Deligne等人的Dynkin情形结果。
We study the bounded derived categories of torus-equivariant coherent sheaves on smooth toric varieties and Deligne-Mumford stacks. We construct and describe full exceptional collections in these categories. We also observe that these categories depend only on the PL homeomorphism type of the corresponding simplicial complex.
Relying on a monadic description of the moduli space of framed sheaves on Hirzebruch surfaces, we construct ADHM data for the Hilbert scheme of points of the total space of the line bundle $\mathcal O(-n)$ on $\mathbb P^1$.
We study Le Potier's strange duality conjecture on a rational surface. We focus on the case involving the moduli space of rank 2 sheaves with trivial first Chern class and second Chern class 2, and the moduli space of 1-dimensional sheaves with determinant $L$ and Euler characteristic 0. We show the conjecture for this case is true under some suitable conditions on $L$, which applies to $L$ ample on any Hirzebruch surface $Σ_e:=\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(e))$ except for $e=1$. When $e=1$, our result applies to $L=aG+bF$ with $b\geq a+[a/2]$, where $F$…
In this paper we construct a full exceptional collection of sheaves on some family of log Del Pezzo surfaces, viewed as a smooth stack. These surfaces can be embedded as a quasi-smooth hypersurfaces into a certain weighted projective space, and they are not toric.
It is proved that any strictly exceptional collection generating the derived category of coherent sheaves on a smooth projective variety X with \rk K_0(X) = \dim X + 1 constists of locally free sheaves up to a common shift.
Structure theorems for exceptional objects and exceptional collections of the bounded derived category of coherent sheaves on del Pezzo surfaces are established by Kuleshov and Orlov. In this paper we propose conjectures which generalize these results to weak del Pezzo surfaces. Unlike del Pezzo surfaces, an exceptional object on a weak del Pezzo surface is not necessarily a shift of a sheaf and is not determined by its class in the Grothendieck group. Our conjectures explain how these complications are taken care of by spherical twists, the categorification of $(-2)$-reflections acting on the…
This is the second of series of papers studyig moduli spaces of a certain class of coherent sheaves, which we call stable perverse coherent sheaves, on the blow-up of a projective surface at a point. The followings are main results of this paper: a) We describe the wall-crossing between moduli spaces caused by twisting of the line bundle associated with the exceptional divisor. b) We give the formula for virtual Hodge numbers of moduli spaces of stable perverse coherent sheaves. Moreover we also give proofs of the followings which we observed in a special case in arXiv:0802.3120: c) The moduli…
Symplectic log Kodaira dimension $-\infty$, Hirzebruch--Jung strings and weighted projective planes
本文研究加权投影平面CP(a,b,c)的辛极小解析，从断开辛除子的视角出发，引入例外间隙概念，建立某些三希尔兹-朱恩字符串配置的托里利型定理，并证明辛仿射刻度的存在性与加权投影平面极小解析的联系。
We investigate combinatorial aspects of exceptional sequences in the derived category of coherent sheaves on certain smooth and complete algebraic surfaces. We show that to any such sequence there is canonically associated a complete toric surface whose torus fixpoints are either smooth or cyclic T-singularities (in the sense of Wahl) of type $\frac{1}{r^2}(1, kr - 1)$. We also show that any exceptional sequence can be transformed by mutation into an exceptional sequence which consists only of objects of rank one.
Let $X$ be a del Pezzo surface. When the degree of $X$ is at least 4, we compute the cohomology of a general sheaf in the moduli space of Gieseker semistable sheaves. We also classify the Chern characters for which the general sheaf in the moduli space is non-special, i.e. has at most one nonzero cohomology group. Our results hold for arbitrary polarizations, slope semistability, and semi-exceptional moduli spaces. When the degree of $X$ is at least 3, we further show our construction of certain vector bundles implies the existence of stable and semistable sheaves with respect to the anti-cano…
We consider the bounded derived category of $S_k$-equivariant coherent sheaves on $(\mathbb{P}^n)^k$. The goal of this paper is to construct in this category a rectangular Lefschetz exceptional collection when this is possible, or a minimal Lefschetz exceptional collection when a rectangular one does not exist. The main results of the paper include the construction of a rectangular Lefschetz exceptional collection in the case $k=3$ and in the case $n=1$ when $\mathrm{gcd}(n+1,k)=1$. We also construct minimal Lefschetz exceptional collection for $n=1$ and even $k$, and for $n=2$ and $k=3$.
Let $\cP=G/P$ be a homogeneous projective variety with $G$ a reductive group and $P$ a parabolic subgroup. In positive characteristic we exhibit for $G$ of low rank a Karoubian complete strongly exceptional poset of locally free sheaves appearing in the Frobenius direct image of the structure sheaf of $G/P$. These sheaves are all defined over $\bbZ$, so by base change provide a Karoubian complete strongly exceptional poset on $\cP$ over $\bbC$, adding to the list of classical results by Beilinson and Kapranov on the Grassmannians and the quadrics over $\bbC$.
We study complete exceptional collections of coherent sheaves over Del Pezzo surfaces, which consist of three blocks such that inside each block all Ext groups between the sheaves are zero. We show that the ranks of all sheaves in such a block are the same and the three ranks corresponding to a complete 3-block exceptional collection satisfy a Markov-type Diophantine equation that is quadratic in each variable. For each Del Pezzo surface, there is a finite number of these equations; the complete list is given. The 3-string braid group acts by mutations on the set of complete 3-block exceptiona…
Let $ϕ:X\rightarrow \mathbb{P}^n$ be a morphism of varieties. Given a hyperplane $H$ in $\mathbb{P}^n$, there is a Gysin map from the compactly supported cohomology of $ϕ^{-1}(H)$ to that of $X$. We give conditions on the degree of the cohomology under which this map is an isomorphism for all but a low-dimensional set of hyperplanes, generalizing results due to Skorobogatov, Benoist, and Poonen-Slavov. Our argument is based on Beilinson's theory of singular supports for étale sheaves.
Let $ξ$ be a stable Chern character on $\mathbb{P}^1 \times \mathbb{P}^1$, and let $M(ξ)$ be the moduli space of Gieseker semistable sheaves on $\mathbb{P}^1 \times \mathbb{P}^1$ with Chern character $ξ$. In this paper, we provide an approach to computing the effective cone of $M(ξ)$ after showing that it is a Mori dream space for all $ξ$. We find Brill-Noether divisors spanning extremal rays of the effective cone using resolutions of the general elements of $M(ξ)$ which are found using the machinery of exceptional bundles. We use this approach to provide many examples of extremal rays in thes…
We work out properties of smooth projective varieties over a (not necessarily algebraically closed) field that admit collections of objects in the bounded derived category of coherent sheaves that are either full exceptional, or numerically exceptional of maximal length. Our main result gives a necessary and sufficient condition on the Neron-Severi lattice for a smooth projective surface with holomorphic Euler characteristic 1 to admit a numerically exceptional collection of maximal length, consisting of line-bundles. As a consequence we determine exactly which complex surfaces with vanishing …
Let $X$ be any rational surface. We construct a tilting bundle $T$ on $X$. Moreover, we can choose $T$ in such way that its endomorphism algebra is quasi-hereditary. In particular, the bounded derived category of coherent sheaves on $X$ is equivalent to the bounded derived category of finitely generated modules over a finite dimensional quasi-hereditary algebra $A$. The construction starts with a full exceptional sequence of line bundles on $X$ and uses universal extensions. If $X$ is any smooth projective variety with a full exceptional sequence of coherent sheaves (or vector bundles, or even…
Let S be a smooth projective surface, K be the canonical class of S and H be an ample divisor such that H.K<0 . In this paper we prove that for any rigid (Ext^1(F,F)=0) semistable sheaf F in the sense of Mumford--Takemoto stability w.r.t. H there exists an exceptional collection (E_1,...,E_n) of sheaves on S such that F can be constructed from {E_i} by a finite number of extensions.
In this note we apply the techniques of the toric systems introduced by Hille-Perling to several problems on smooth projective surfaces: We showed that the existence of full exceptional collection of line bundles implies the rationality for small Picard rank surfaces; we proved equivalences of several notions of cyclic strong exceptional collection of line bundles; we also proposed a partial solution to a conjecture on exceptional sheaves on weak del Pezzo surfaces.
Higher-dimensional O'Grady's resolutions and their exceptional locus
本文研究 K3 曲面上 Mukai 向量为 $2(1,0,1-n)$ 的半稳定层模空间 $M_n$，分析 O'Grady 构造的显式奇点解消，并全局刻画其例外轨迹，为探讨范畴 crepant 解消奠定基础。
We construct quasi-phantom admissible subcategories in the derived category of coherent sheaves on the Beauville surface $S$. These quasi-phantoms subcategories appear as right orthogonals to subcategories generated by exceptional collections of maximal possible length 4 on $S$. We prove that there are exactly 6 exceptional collections consisting of line bundles (up to a twist) and these collections are spires of two helices.
In this paper we deal with semistable sheaves which can be represented as the cokernel of an injective (or as the kernel of a surjective) morphism $E_1\otimes\CC^m\longrightarrow E_2\otimes\CC^n$ , where $E_1$ and $E_2$ are exceptional bundles. To each such a sheaf we assign the linear map $\CC^m\otimes\hom^*(E_1,E_2)\longrightarrow \CC^n$. We obtain sufficient conditions on the topological invariants of the sheaves for the moduli space of the sheaves to be isomorphic to the quotient of an open subset in $\PP{\cal L}(\CC^m\otimes\hom^*(E_1,E_2),\CC^n)$ under the action of SL$(\CC^m)\times$SL$(…
Towards homological projective duality for $\mathrm{Gr}(2, 2n)$
本文针对偶维向量空间的格拉斯曼簇$\mathrm{Gr}(2,V)$，构造了普法夫簇上$\mathbb{P}^1$纤维化簇$X_1$的凝聚层例外集合，猜想其生成的导出子范畴是该格拉斯曼簇的同调射影对偶范畴，推进了同调射影对偶理论研究。
Vafa-Witten invariants from exceptional collections
针对容许强例外层集的曲面，提出利用带势箭图表示的导出范畴计算Vafa-Witten不变量的通用方法，验证了结果与已有计算一致，并给出相关模函数和黑洞存在性的推论。
Mutations of Numerically Exceptional Collections on Surfaces
本文研究了曲面上数值例外集合的变异，证明在有理曲面上辫群在极大数值例外集合上的传递性，并将其提升至导出范畴，推广了关于 del Pezzo 曲面的已有结果。
Moduli spaces of semistable sheaves on a K3 or abelian surface with respect to a general ample divisor are shown to be locally factorial, with the exception of symmetric products of a K3 or abelian surface and the class of moduli spaces found by O'Grady. Consequently, since singular moduli space that do not belong to these exceptional cases have singularities in codimension $\geq4$ they do no admit projective symplectic resolutions.
In this paper, we apply the idea of T-duality to projective spaces. From a connection on a line bundle on $\mathbb P^n$, a Lagrangian in the mirror Landau-Ginzburg model is constructed. Under this correspondence, the full strong exceptional collection $\mathcal O_{\mathbb P^n}(-n-1),...,\mathcal O_{\mathbb P^n}(-1)$ is mapped to standard Lagrangians in the sense of \cite{nz}. Passing to constructible sheaves, we explicitly compute the quiver structure of these Lagrangians, and find that they match the quiver structure of this exceptional collection of $\mathbb P^n$. In this way, T-duality prov…
Recently de Thanhoffer de Völcsey and Van den Bergh classified the Euler forms on a free abelian group of rank 4 having the properties of the Euler form of a smooth projective surface. There are two types of solutions: one corresponding to $\mathbb{P}^1\times\mathbb{P}^1$ (and noncommutative quadrics), and an infinite family indexed by the natural numbers. For $m=0,1$ there are commutative and noncommutative surfaces having this Euler form, whilst for $m\geq 2$ there are no commutative surfaces. In this paper we construct sheaves of maximal orders on surfaces having these Euler forms, giving a…
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