The Severi problem for Hirzebruch surfaces

Exceptional sheaves on the Hirzebruch surface $\mathbb{F}_2$

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.

Rational cuspidal curves with four cusps on Hirzebruch surfaces

The purpose of this article is to shed light on the question of how many and what kind of cusps a rational cuspidal curve on a Hirzebruch surface can have. We use birational transformations to construct rational cuspidal curves with four cusps on the Hirzebruch surfaces and find associated results for these curves.

Stable ample 2-vector bundles on Hirzebruch surfaces

We discuss stability conditions for all rank-2 ample vector bundles on Hirzebruch surfaces with the second Chern class less than 7.

Korff $F$-signatures of Hirzebruch surfaces

M. Von Korff introduced the $F$-signature of a normal projective variety, and computed the $F$-signature of the product of two projective lines $\mathbb{P}^1\times \mathbb{P}^1$. In this paper, we compute $F$-signatures of Hirzebruch surfaces.

理性曲面上的线性系统

Linear systems on blow-ups of Hirzebruch surfaces

本文提出并证明了Hirzebruch曲面上关于线性系统的猜想，适用于特定点数的吹胀情况。

Hirzebruch-Kummer covers of algebraic surfaces

The aim of this paper is to show that using some natural curve arrangements in algebraic surfaces and Hirzebruch-Kummer covers one cannot construct new examples of ball-quotients, i.e., minimal smooth complex projective surfaces of general type satisfying equality in the Bogomolov-Miyaoka-Yau inequality.

On the number of cusps on cuspidal curves on Hirzebruch surfaces

In this article we give an upper bound for the number of cusps on a cuspidal curve on a Hirzebruch surface. We adapt the results that have been found for a similar question asked for cuspidal curves on the projective plane, and restate the results in this new setting.

Classification of globally F-regular $F$-sandwiches of Hirzebruch surfaces

Let $X$ be a smooth variety over an algebraically closed field of positive characteristic. An $F$-sandwich of $X$ is a normal variety $Y$ through which the relative Frobenius morphism of $X$ factors as $F:X\rightarrow Y \rightarrow X$. In this paper, we give a classification of globally F-regular $F$-sandwiches of Hirzebruch surfaces.

Hirzebruch曲面上的Segre不变量与Brill-Noether理论

The Segre invariant and Brill-Noether theory on Hirzebruch surfaces

本文研究Hirzebruch曲面上特定秩向量丛的Segre不变量，确定其取值、计算整体截面数，并将结果应用于Brill-Noether理论。

Vector bundles on Hirzebruch surfaces whose twists by a non-ample line bundle have natural cohomology

Here we study vector bundles $E$ on the Hirzebruch surface $F_e$ such that their twists by a spanned, but not ample, line bundle $M = \mathcal {O}_{F_e}(h+ef)$ have natural cohomology, i.e. $h^0(F_e,E(tM)) >0$ implies $h^1(F_e,E(tM)) = 0$.

Brill-Noether theorems and globally generated vector bundles on Hirzebruch surfaces

In this paper, we show that the cohomology of a general stable bundle on a Hirzebruch surface is determined by the Euler characteristic provided that the first Chern class satisfies necessary intersection conditions. More generally, we compute the Betti numbers of a general stable bundle. We also show that a general stable bundle on a Hirzebruch surface has a special resolution generalizing the Gaeta resolution on the projective plane. As a consequence of these results, we classify Chern characters such that the general stable bundle is globally generated.

Hirzebruch曲面爆破上的幻影子范畴

Phantom subcategories on blow-ups of Hirzebruch surfaces

本文证明Hirzebruch曲面在一般点爆破后存在幻影子范畴，覆盖所有n≥3的情况。

Hirzebruch 曲面上超椭圆曲线模空间的稳定上同调

Stable cohomology of moduli spaces of hyperelliptic curves on Hirzebruch surfaces

本文计算了 Hirzebruch 曲面上固定亏格超椭圆曲线模空间的稳定上同调，并用带标记点的非嵌入模空间进行描述。

Hirzebruch曲面一般爆胀上线丛的正性

Positivity of line bundles on general blow ups of Hirzebruch surfaces

本文研究Hirzebruch曲面一般爆胀上线丛的正性，针对丰富性、整体生成性、非常丰富性和k-非常丰富性分别给出充分的数值条件。

Hirzebruch曲面的非正与无穷远负除子赋值

Non-positive and negative at infinity divisorial valuations of Hirzebruch surfaces

本文研究由Hirzebruch曲面除子赋值定义的有理曲面，引入非正性与无穷远负性概念，证明其局部与全局等价条件，并在非正无穷远情形下显式给出曲线锥的极值射线。

Hirzebruch曲面上的曲线与半稳定性

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。

Hilbert scheme of some threefold scrolls over the Hirzebruch surface F_1

Hilbert schemes of suitable smooth, projective manifolds of low degree which are 3-fold scrolls over the Hirzebruch surface F_1 are studied. An irreducible component of the Hilbert scheme parametrizing such varieties is shown to be generically smooth of the expected dimension and the general point of such a component is described.

Monads for framed sheaves on Hirzebruch surfaces

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.

Hirzebruch面上稳定性条件空间中的墙结构

Wall in the stability space of the gluing stability conditions on Hirzebruch surfaces

本文研究Hirzebruch面上稳定性条件空间的墙结构，通过粘合构造分析几何室与墙的交集，并确定半稳定对象的模空间。

On fundamental groups related to the Hirzebruch surface F_1

Given a projective surface and a generic projection to the plane, the braid monodromy factorization (and thus, the braid monodromy type) of the complement of its branch curve is one of the most important topological invariants, stable on deformations. From this factorization, one can compute the fundamental group of the complement of the branch curve, either in C^2 or in CP^2. In this article, we show that these groups, for the Hirzebruch surface F_{1,(a,b)}, are almost-solvable. That is - they are an extension of a solvable group, which strengthen the conjecture on degeneratable surfaces.

Hirzebruch曲面的orbifold退化

Orbifold Degenerations of Hirzebruch Surfaces

该研究探讨Hirzebruch曲面的orbifold退化，通过证明其为环面曲面的部分光滑化，结合三维极小模型程序等，分类了不同k值下的中心纤维及对应奇点，补充了反典范除子丰富情形的相关工作。

Orbifold kähler groups and the shafarevich conjecture for hirzebruch's covering surfaces with equal weights

This article is devoted to examples of (orbifold) Kähler groups from the perspective of the so-called Shafarevich conjecture on holomorphic convexity. It aims at pointing out that every quasi-projective complex manifold with an 'interesting' fundamental group gives rise to interesting instances of this long-standing open question. Complements of line arrangements are one of the better known classes of quasi-projective complex surfaces with an interesting fundamental group. We solve the corresponding instance of the Shafarevich conjecture partially giving a proof that the universal covering sur…

Del Pezzo Surfaces, Rigid Line Configurations and Hirzebruch-Kummer Coverings

We prove the equisingular rigidity of the singular Hirzebruch-Kummer coverings $X(n, \mathcal{L})$ of the projective plane branched on line configurations $\mathcal{L}$, satisfying some technical condition. In the case, $\mathcal{L}$ = the complete quadrangle, we give explicit equations of the Hirzebruch-Kummer covering $S_n$ (=the minimal desingularisation of $X(n, \mathcal{L})$) in a product of four Fermat curves of degree n. Since $S_n$ is the $(\mathbb{Z}/n)^5$ covering of the Del Pezzo surface $Y_5$ of degree 5 branched on the 10 lines, these equations are derived from explicit equations …

On real anti-bicanonical curves with one double point on the $4$-th real Hirzebruch surface. II

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.

Prioritary omalous bundles on Hirzebruch surfaces

An irreducible algebraic stack is called \emph{unirational} if there exists a surjective morphism, representable by algebraic spaces, from a rational variety to an open substack. We prove unirationality of the stack of prioritary omalous bundles on Hirzebruch surfaces, which implies also the unirationality of the moduli space of omalous $H$-stable bundles for any ample line bundle $H$ on a Hirzebruch surface. To this end, we find an explicit description of the duals of omalous rank-two bundles with a vanishing condition in terms of monads. Since these bundles are prioritary, we conclude that t…

Hirzebruch曲面上三分量曲线的代数例外集

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。

Gersten-Witt Complex of Hirzebruch Surfaces

Gersten-Witt cohomologies of Hirzebruch surfaces are computed.

Trigonal Morsifications on Hirzebruch Surfaces with an appendix by E. Shustin

In this paper we obtain a classification of rigid isotopy classes of totally reducible trigonal curves lying on a Hirzebruch surface $Σ_n$, and having a maximal number of non-degenerated double points. Such curves correspond to morsifications of a totally real semiquasihomogeneous singularity of weight $(3,3n)$ (the union of three smooth real branches intersecting each other with multiplicity $n$). We obtain this classification by studying combinatorial properties of dessins. In the appendix, we prove that any morsification of a totally real semiquasihomogeneous singularity of weight $(3,3n)$ …

The boundary of the Milnor fiber of Hirzebruch surface singularities

We give the first (as far as we know) complete description of the boundary of the Milnor fiber for some non-isolated singular germs of surfaces in ${\bf C}^3$. We study irreducible (i.e. $gcd (m,k,l) = 1$) non-isolated (i.e. $1 \leq k < l$) Hirzebruch hypersurface singularities in ${\bf C}^3$ given by the equation $z^m - x^ky^l = 0$. We show that the boundary $L$ of the Milnor fiber is always a Seifert manifold and we give an explicit description of the Seifert structure. From it, we deduce that : 1) $L$ is never diffeomorphic to the boundary of the normalization. 2) $L$ is a lens space iff $m…

On families of rank-2 uniform bundles on Hirzebruch surfaces and Hilbert schemes of their scrolls

Several families of rank-two vector bundles on Hirzebruch surfaces are shown to consist of all very ample, uniform bundles. Under suitable numerical assumptions, the projectivization of these bundles, embedded by their tautological line bundles as linear scrolls, are shown to correspond to smooth points of components of their Hilbert scheme, the latter having the expected dimension. If e=0,1 the scrolls fill up the entire component of the Hilbert scheme, while for e=2 the scrolls exhaust a subvariety of codimension 1.

Maximally inflected real trigonal curves on Hirzebruch surfaces

In 2014 A. Degtyarev, I. Itenberg and the author gave a description, up to fiberwise equivariant deformations, of maximally inflected real trigonal curves of type~I (over a base $ B $ of an arbitrary genus) in terms of the combinatorics of sufficiently simple graphs and for $ B=\mathbb{P}^1 $ obtained a complete classification of such curves. In this paper, the mentioned results are extended to maximally inflected real trigonal curves of type~II over $ B=\mathbb{P}^1 $. As usual, the results for such curves are extended to real Jacobian elliptic surfaces with all singular fibers real.

Hirzebruch 曲面的亏格 0 对数与热带固定定义域计数

Genus 0 logarithmic and tropical fixed-domain counts for Hirzebruch surfaces

本文研究环簇的虚对数 Tevelev 度，证明亏格 0 对应定理，并利用热带曲线枚举给出 Hirzebruch 曲面情形的闭式计数公式。

Hirzebruch曲面上有理曲面的曲线锥与Cox环

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。

关于Hirzebruch曲面上秩2向量丛的模空间的注记

A Note On the moduli space of rank two vector bundles on Hirzebruch surfaces

本文研究Hirzebruch曲面$\text{F}_e$上秩2向量丛的Segre不变量，给出其取值必要条件、层维数界，应用于Brill-Noether理论并分析极化变化下模空间的差异。

平面与 Hirzebruch 曲面上光滑曲线的 Brill--Noether 理论

Brill--Noether theory of smooth curves in the plane and on Hirzebruch surfaces

本文研究了一般光滑平面曲线及 Hirzebruch 曲面上固定类曲线的 Brill--Noether 理论，通过固定线丛及其扭变的分裂类型细化分层，确定了各局部闭合层的维度并证明了基域特征为零时的光滑性。

Moduli of noncommutative Hirzebruch surfaces

We introduce three non-compact moduli stacks parametrizing noncommutative deformations of Hirzebruch surfaces; the first is the moduli stack of locally free sheaf bimodules of rank 2, which appears in the definition of noncommutative $\mathbb{P}^1$-bundle in the sense of Van den Bergh arXiv:math/0102005, the second is the moduli stack of relations of a quiver in the sense of arXiv:1411.7770, and the third is the moduli stack of quadruples consisting of an elliptic curve and three line bundles on it. The main result of this paper shows that they are naturally birational to each other. We also g…

Hirzebruch曲面上无穷远处非正赋值的Seshadri型常数与Newton-Okounkov体

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 Quantitative Behavior of Asymptotic Syzygies for Hirzebruch Surfaces

The goal of this note is to quantitatively study the behavior of asymptotic syzygies for certain toric surfaces, including Hirzebruch surfaces. In particular, we show that the asymptotic linear syzygies of Hirzebruch surfaces embedded by $\mathcal{O}(d,2)$ conform to Ein, Erman, and Lazarsfeld's normality heuristic. We also show that the higher degree asymptotic syzygies are not asymptotically normally distributed.

Deformation of tropical Hirzebruch surfaces and enumerative geometry

We illustrate the use of tropical methods by generalizing a formula due to Abramovich and Bertram, extended later by Vakil. Namely, we exhibit relations between enumerative invariants of the Hirzebruch surfaces $Σ_n$ and $Σ_{n+2}$, obtained by deforming the first surface to the latter. Our strategy involves a tropical counterpart of deformations of Hirzebruch surfaces, and tropical enumerative geometry on a tropical surface in three-space.

Topological Formulae for the Zeroth Cohomology of Line Bundles on del Pezzo and Hirzebruch Surfaces

We show that the zeroth cohomology of effective line bundles on del Pezzo and Hirzebruch surfaces can always be computed in terms of a topological index.

Characterization of Ulrich bundles on Hirzebruch surfaces

In this work we characterize Ulrich bundles of any rank on polarized rational ruled surfaces over $\mathbb{P}^1$. We show that every Ulrich bundle admits a resolution in terms of line bundles. Conversely, given an injective map between suitable totally decomposed vector bundles, we show that its cokernel is Ulrich if it satisfies a vanishing in cohomology. As a consequence we obtain, once we fix a polarization, the existence of Ulrich bundles for any admissible rank and first Chern class. Moreover we show the existence of stable Ulrich bundles for certain pairs $(\textrm{rk}(E),c_1(E))$ and wi…

New examples of Weierstrass semigroups associated with a double covering of a curve on a Hirzebruch surface of degree one

Let $φ:Σ_1\longrightarrow \mathbb{P}^2$ be a blow up at a point on $\mathbb{P}^2$. Let $C$ be the proper transform of a smooth plane curve of degree $d\geq 4$ by $φ$, and let $P$ be a point on $C$. Let $π:\tilde{C}\longrightarrow C$ be a double covering branched along the reduced divisor on $C$ obtained as the intersection of $C$ and a reduced divisor in $|-2K_{Σ_1}|$ containing $P$. In this paper, we investigate the Weierstrass semigroup $H(\tilde{P})$ at the ramification point $\tilde{P}$ of $π$ over $P$, in the case where the intersection multiplicity at $φ(P)$ of $φ(C)$ and the tangent lin…

Counting curves on Hirzebruch surfaces: tropical geometry and the Fock space

We study the stationary descendant Gromov-Witten theory of toric surfaces by combining and extending a range of techniques - tropical curves, floor diagrams, and Fock spaces. A correspondence theorem is established between tropical curves and descendant invariants on toric surfaces using maximal toric degenerations. An intermediate degeneration is then shown to give rise to floor diagrams, giving a geometric interpretation of this well-known bookkeeping tool in tropical geometry. In the process, we extend floor diagram techniques to include descendants in arbitrary genus. These floor diagrams …

Topological obstructions for rational cuspidal curves in Hirzebruch surfaces

We study rational cuspidal curves in Hirzebruch surfaces. We provide two obstructions for the existence of rational cuspidal curves in Hirzebruch surfaces with prescribed types of singular points. The first result comes from Heegaard--Floer theory and is a generalization of a result by Livingston and the first author. The second criterion is obtained by comparing the spectrum of a suitably defined link at infinity of a curve with spectra of its singular points.

Counting generic genus-0 curves on Hirzebruch surfaces

Hirzebruch surfaces provide an excellent example to underline the fact that in general symplectic manifolds, Gromov-Witten invariants might well count curves in the boundary components of the moduli space. We use this example to explain in detail that the counting argument given by Batyrev in "Quantum cohomology rings of toric manifolds" (Asterisque 218:9-34, 1993) does not work.

商Deligne-Mumford栈的轨形Hirzebruch-Riemann-Roch定理与K3曲面上的等变模空间理论

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。

The double Gromov-Witten invariants of Hirzebruch surfaces are piecewise polynomial

We define the double Gromov-Witten invariants of Hirzebruch surfaces in analogy with double Hurwitz numbers, and we prove that they satisfy a piecewise polynomiality property analogous to their 1-dimensional counterpart. Furthermore we show that each polynomial piece is either even or odd, and we compute its degree. Our methods combine floor diagrams and Ehrhart theory.

Hirzebruch曲面爆破上的Seshadri常数

Seshadri constants on blow-ups of Hirzebruch surfaces

本文研究 Hirzebruch 曲面多点爆破上丰富线丛的 Seshadri 常数，给出 $r\le e-1$ 时任一点及 $r=e,e+1$ 时极一般点处的计算结果，并讨论相关曲线线性系统猜想。

映射到希策布施表面的曲线的卷积不变量

The Scrollar Invariants of Curves Mapping to a Hirzebruch Surface

本文研究了映射到希策布施表面的曲线的卷积不变量，并给出了具有固定类和奇点位的节点曲线的存在定理。

Analytic torsion of Hirzebruch surfaces

Using different forms of the arithmetic Riemann-Roch theorem and the computations of Bott-Chern secondary classes, we compute the analytic torsion and the height of Hirzebruch surfaces.

Existence of semistable sheaves on Hirzebruch surfaces

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…

The effect of points fattening on Hirzebruch surfaces

The purpose of this note is to study initial sequences of zero-dimensional subschemes of Hirzebruch surfaces and classify subschemes whose initial sequence has the minimal possible growth.

Angle deformation of Kähler-Einstein edge metrics on Hirzebruch surfaces

We construct a family of Kähler-Einstein edge metrics on all Hirzebruch surfaces using the Calabi ansatz and study their angle deformation. This allows us to verify in some special cases a conjecture of Cheltsov-Rubinstein that predicts convergence towards a non-compact Calabi-Yau fibration in the small angle limit. We also give an example of a Kähler-Einstein edge metric whose edge singularity is rigid, answering a question posed by Cheltsov.

Refined curve counting with tropical geometry

The Severi degree is the degree of the Severi variety parametrizing plane curves of degree d with delta nodes. Recently, Göttsche and Shende gave two refinements of Severi degrees, polynomials in a variable y, which are conjecturally equal, for large d. At y = 1, one of the refinements, the relative Severi degree, specializes to the (non-relative) Severi degree. We give a tropical description of the refined Severi degrees, in terms of a refined tropical curve count for all toric surfaces. We also refine the equivalent count of floor diagrams for Hirzebruch and rational ruled surfaces. Our desc…