Stable maps of genus zero to flag spaces

A remark on the genus of curves in $\mathbf P^4$

Let $C$ be an irreducible, reduced, non-degenerate curve, of arithmetic genus $g$ and degree $d$, in the projective space $\mathbf P^4$ over the complex field. Assume that $C$ satisfies the following {\it flag condition of type $(s,t)$}: {$C$ does not lie on any surface of degree $<s$, and on any hypersurface of degree $<t$}. Improving previous results, in the present paper we exhibit a Castelnuovo-Halphen type bound for $g$, under the assumption $s\leq t^2-t$ and $d\gg t$. In the range $t^2-2t+3\leq s\leq t^2-t$, $d\gg t$, we are able to give some information on the extremal curves. They are …

K-theoretic J-functions of type A flag varieties

The J-function in Gromov-Witten theory is a generating function for one-point genus zero Gromov-Witten invariants with descendants. Here we give formulas for the quantum K-theoretic J-functions of type A flag manifolds and conjectural formulas for other types. Some K-theoretic tools for computation are also provided. As an application, we prove the quantum K-theoretic J-function version of the abelian-nonabelian correspondence for Grassmannians and products of projective space.

Towards the cohomology of moduli spaces of higher genus stable maps

We prove that the orbifold desingularization of the moduli space of stable maps of genus g = 1 recently constructed by Vakil and Zinger has vanishing rational cohomology groups in odd degree k < 10.

Divisor classes and the virtual canonical bundle for genus 0 maps

Some divisor class relations are proved and used to compute the Cartier divisor class of the virtual canonical bundle for genus 0 maps to a smooth target.

A splitting of the virtual class for genus one stable maps

Moduli spaces of stable maps to a smooth projective variety typically have several components. We express the virtual class of the moduli space of genus one stable maps to a smooth projective variety as a sum of virtual classes of the components. The key ingredient is a generalised functoriality result for virtual classes. We show that the natural maps from 'ghost' components of the genus one moduli space to moduli spaces of genus zero stable maps satisfy the strong push forward property. As a consequence, we give a cycle-level formula which relates standard and reduced genus one Gromov--Witte…

On the irreducibility of the space of genus zero stable log maps to wonderful compactifications

In this paper, we prove the moduli spaces of genus zero stable log maps to a large class of wonderful compactifications are irreducible and unirational.

Universal cohomological expressions for singularities in families of genus 0 stable maps

We consider families of curve-to-curve maps that have no singularities except those of genus 0 stable maps and that satisfy a versality condition at each singularity. We provide a universal expression for the cohomology class Poincaré dual to the locus of any given singularity. Our expressions hold for any family of curve-to-curve maps satisfying the above properties.

Trigonometric weight functions as K-theoretic stable envelope maps for the cotangent bundle of a flag variety

We consider the cotangent bundle $T^*F_λ$ of a $GL_n$ partial flag variety, $λ=(λ_1,...,λ_N)$, $|λ|=\sum_iλ_i=n$, and the torus $T=(\C^\times)^{n+1}$ equivariant K-theory algebra $K_T(T^*F_λ)$. We introduce K-theoretic stable envelope maps $\Stab_σ: \oplus_{|λ|=n} K_T((T^*F_λ)^T)\to\oplus_{|λ|=n}K_T(T^*F_λ)$, where $σ\in S_n$. Using these maps we define a quantum loop algebra action on $\oplus_{|λ|=n}K_T(T^*F_λ)$. We describe the associated Bethe algebra $B^q(K_T(T^*F_λ))$ by generators and relations in terms of a discrete Wronski map. We prove that the limiting Bethe algebra $B^q(K_T(T^*F_λ))…

Genus-One Stable Maps, Local Equations, and Vakil-Zinger's desingularization

We describe an algebro-geometric approach to Vakil-Zinger's desingularization of the main component of the moduli of genus one stable maps to projective space. The new approach provides complete local structural results for this moduli space as well as for the desingularization of the entire moduli space and should fully extend to higher genera.

双有理映射、PBW退化旗与偏序集多面体

Birational maps, PBW degenerate flags and poset polytopes

本研究将射影空间与格拉斯曼流形间双有理映射的图闭包结果推广至PBW退化旗簇，分析其组合、代数与几何性质，验证了环面退化的普适性，并通过箭图表示刻画旗簇图闭包。

Quantum K theory for flag varieties

Forgetting a subspace from a partial flag yields another partial flag composed of fewer subspaces. This induces a forgetful map $π: X \to X'$ between the corresponding flag varieties. We prove here that, for a degree large enough, the variety associated with degree d stable maps sending their marked points within Schubert varieties $X_i$ of $X$ is a rationally connected fibration over its image, which parametrizes degree $π_* d$ stable maps sending their marked points within the Schubert varieties $π(X_i)$ of $X'$. The Euler characteristic of these varieties are quantum $K$-invariants. Our res…

Four dimensional Fano quiver flag zero loci (with an appendix by T. Coates, E. Kalashnikov, and A. Kasprzyk)

Quiver flag zero loci are subvarieties of quiver flag varieties cut out by sections of homogeneous vector bundles. We prove the Abelian/non-Abelian Correspondence in this context: this allows us to compute genus zero Gromov--Witten invariants of quiver flag zero loci. We determine the ample cone of a quiver flag variety, disproving a conjecture of Craw. In the Appendices, which are joint work with Tom Coates and Alexander Kasprzyk, we use these results to find four-dimensional Fano manifolds that occur as quiver flag zero loci in ambient spaces of dimension up to 8, and compute their quantum p…

正特征下平向量丛的完全旗

Complete flags on flat vector bundles in positive characteristic

本文研究正特征下平向量丛的相关问题，证明亏格g≤1等价于X上任意平向量丛存在完全旗，并探索该结果的更广泛推广。

Moduli of stable maps in genus one and logarithmic geometry I

This is the first in a pair of papers developing a framework for the application of logarithmic structures in the study of singular curves of genus $1$. We construct a smooth and proper moduli space dominating the main component of Kontsevich's space of stable genus $1$ maps to projective space. A variation on this theme furnishes a modular interpretation for Vakil and Zinger's famous desingularization of the Kontsevich space of maps in genus $1$. Our methods also lead to smooth and proper moduli spaces of pointed genus $1$ quasimaps to projective space. Finally, we present an application to t…

An Invitation to the Local Structures of Moduli of Genus one stable maps

This informal note provides some elementary examples to motivate the local structural results of [1] on the moduli space of genus one stable maps to projective space. The hope is that these examples will be helpful for graduate students to learn this important subject.

Real aspects of the moduli space of genus zero stable maps

We show that the moduli space of genus zero stable maps is a real projective variety if the target space is a smooth convex real projective variety. We show that evaluation maps, forgetful maps are real morphisms. We analyze the real part of the moduli space.

genus 0 映射到旗 manifold 的动机类

The motivic class of the space of genus $0$ maps to the flag variety

本文研究了 genus 0 映射到旗 manifold 的动机类，并在正性条件下证明其在 K_0(Var) 中的类为 GL_n×A^a。

Moduli of stable maps in genus one and logarithmic geometry II

This is the second in a pair of papers developing a framework to apply logarithmic methods in the study of singular curves of genus $1$. This volume focuses on logarithmic Gromov--Witten theory and tropical geometry. We construct a logarithmically nonsingular moduli space of genus $1$ curves mapping to any toric variety. The space is a birational modification of the principal component of the Abramovich--Chen--Gross--Siebert space of logarithmic stable maps and produces an enumerative genus $1$ curve counting theory. We describe the non-archimedean analytic skeleton of this moduli space and, a…

Explicit Solution By Radicals, Gonal Maps and Plane Models of Algebraic Curves of Genus 5 or 6

We give explicit computational algorithms to construct minimal degree (always $\le 4$) ramified covers of $\Prj^1$ for algebraic curves of genus 5 and 6. This completes the work of Schicho and Sevilla (who dealt with the $g \le 4$ case) on constructing radical parametrisations of arbitrary genus $g$ curves. Zariski showed that this is impossible for the general curve of genus $\ge 7$. We also construct minimal degree birational plane models and show how the existence of degree 6 plane models for genus 6 curves is related to the gonality and geometric type of a certain auxiliary surface.

Spaces of quasi-maps into the flag varieties and their applications

Given a projective variety X and a smooth projective curve C one may consider the moduli space of maps C --> X. This space admits certain compactification whose points are called quasi-maps. In the last decade it has been discovered that in the case when X is a (partial) flag variety of a semi-simple algebraic group G (or, more generally, of any symmetrizable Kac-Moody Lie algebra) these compactifications play an important role in such fields as geometric representation theory, geometric Langlands correspondence, geometry and topology of moduli spaces of G-bundles on algebraic surfaces, 4-dime…

Composite Genus One Belyi Maps

Motivated by a demand for explicit genus 1 Belyi maps from theoretical physics, we give an efficient method of explicitly computing genus one Belyi maps by (1) composing covering maps from elliptic curves to the Riemann sphere with simpler (univariate) genus zero Belyi maps, as well as by (2) composing further with isogenies between elliptic curves. This gives many new explicit dessins on the doubly periodic plane, including several which have been realized in the physics literature as so-called brane-tilings in the context of quiver gauge theories.

齐次射影簇之间正则映射的不存在性

Nonexistence of regular maps between homogeneous projective varieties

本文研究部分旗簇间非恒定正则映射的不存在性，证明部分非完全旗簇到完全旗簇不存在此类映射，并提出一般猜想。

Birational self-maps of threefolds of (un)-bounded genus or gonality

We study the complexity of birational self-maps of a projective threefold $X$ by looking at the birational type of surfaces contracted. These surfaces are birational to the product of the projective line with a smooth projective curve. We prove that the genus of the curves occuring is unbounded if and only if $X$ is birational to a conic bundle or a fibration into cubic surfaces. Similarly, we prove that the gonality of the curves is unbounded if and only if $X$ is birational to a conic bundle.

Intersection theoretic properties on the moduli space of genus 0 stable maps to a semipositive symplectic 4-manifold

We characterize transversality, non-transversality properties on the moduli space of genus 0 stable maps to a semipositive symplectic manifold of dimension 4, when GW([point],...,[point]) is enumerative. In particular, we show that the intersection theoretic property depends on the existence of a critical point on a stable map.

Alternate compactifications of the moduli space of genus one maps

We extend the definition of an m-stable curve introduced by Smyth to the setting of maps to a projective variety X, generalizing the definition of a Kontsevich stable map in genus one. We prove that the moduli problem of n-pointed m-stable genus one maps of class β is representable by a proper Deligne-Mumford stack \sMbar_{1,n}^m(X, β) over Spec Z[1/6]. For X = P^r, we explicitly describe all of the irreducible components of \sMbar_{1,n}(P^r,d) and \sMbar_{1,n}^m(P^r,d), and in particular deduce that \sMbar_{1,n}^m(P^r,d) is irreducible for m >= min(r,d) + n. We show that \sMbar_{1,n}^m(P^r,d)…

Equivalence of Mirror Families Constructed from Toric Degenerations of Flag Varieties

Batyrev (et. al.) constructed a family of Calabi-Yau varieties using small toric degenerations of the full flag variety G/B. They conjecture this family to be mirror to generic anti-canonical hypersurfaces in G/B. Recently Alexeev and Brion, as a part of their work on toric degenerations of spherical varieties, have constructed many degenerations of G/B. For any such degeneration we construct a family of varieties, which we prove coincides with Batyrev's in the small case. We prove that any two such families are birational, thus proving that mirror families are independent of the choice of deg…

亏格二稳定映射、局部方程与模消解

Genus Two Stable Maps, Local Equations and Modular Resolutions

本研究通过对参数化亏格二预稳定对的阿廷叠进行模胀开构造，局部对角化相关稳定映射模空间的导出对象，消解主分支奇点并使全空间仅含正规交叉奇点，方法有望推广至高亏格。

On the genus of birational maps between 3-folds

In this note we present two equivalent definitions for the genus of a birational map X --> Y between smooth complex projective 3-folds. The first one is the definition introduced in 1973 by M. A. Frumkin, the second one was recently suggested to me by S. Cantat. By focusing first on proving that these two definitions are equivalent, one can obtain all the results of the paper of Frumkin in a much shorter way. In particular, the genus of an automorphism of $\mathbb{C}^3$, view as a birational self-map of the projective space, will easily be proved to be 0.

Real Gromov-Witten invariants on the moduli space of genus 0 stable maps to a smooth rational projective space

We characterize transversality, non-transversality properties on the moduli space of genus 0 stable maps to a rational projective surface. If a target space is equipped with a real structure, i.e, anti-holomorphic involution, then the results have real enumerative applications. Firstly, we can define a real version of Gromov-Witten invariants. Secondly, we can prove the invariance of Welschinger's invariant in algebraic geometric category.

Stable maps of genus zero in the space of stable vector bundles on a curve

Let $X$ be a smooth projective curve with genus $g\geq3$. Let $\mathcal{N}$ be the moduli space of stable rank two vector bundles on $X$ with a fixed determinant $\mathcal{O}_X(-x)$ for $x\in X$. In this paper, as a generalization of Kiem and Castravet's works, we study the stable maps in $\mathcal{N}$ with genus $0$ and degree $3$. Let $P$ be a natural closed subvariety of $\mathcal{N}$ which parametrizes stable vector bundles with a fixed subbundle $L^{-1}(-x)$ for a line bundle $L$ on $X$. We describe the stable map space $\mathbf{M}_0(P,3)$. It turns out that the space $\mathbf{M}_0(P,3)$ …

Intersection Numbers on the Moduli Spaces of Stable Maps in Genus 0

Let $V$ be a smooth, projective, convex variety. We define tautological $ψ$ and $κ$ classes on the moduli space of stable maps $\M_{0,n}(V)$, give a (graphical) presentation for these classes in terms of boundary strata, derive differential equations for the generating functions of the Gromov-Witten invariants of $V$ twisted by these tautological classes, and prove that these intersection numbers are completely determined by the Gromov-Witten invariants of $V$. This results in families of Frobenius manifold structures on the cohomology ring of $V$ which includes the quantum cohomology as a spe…

Moduli of stable sheaves supported on curves of genus three contained in a quadric surface

We study the moduli space of stable sheaves of Euler characteristic 1 supported on curves of arithmetic genus 3 contained in a smooth quadric surface. We show that this moduli space is rational. We compute its Betti numbers by studying the variation of the moduli spaces of alpha-semi-stable pairs. We classify the stable sheaves using locally free resolutions or extensions. We give a global description: the moduli space is obtained from a certain flag Hilbert scheme by performing two flips followed by a blow-down.

Genus One Belyi Maps by Quadratic Correspondences

We present a method of obtaining a Belyi map on an elliptic curve from that on the Riemann sphere. This is done by writing the former as a radical of the latter, which we call a quadratic correspondence, with the radical determining the elliptic curve. With a host of examples of various degrees we demonstrate that the correspondence is an efficient way of obtaining genus one Belyi maps. As applications, we find the Belyi maps for the dessins d'enfant which have arisen as brane-tilings in the physics community, including ones, such as the so-called suspended pinched point, which have been a sta…

The connectedness of the moduli space of maps to homogeneous spaces

We prove the connectedness of the moduli space of maps (of fixed genus and homology class) to the homogeneous space G/P by degeneration via the maximal torus action. In the genus 0 case, the irreducibility of the moduli of maps is a direct consequence of connectedness. An analysis of a related Bialynicki-Birula stratification of the map space yields a rationality result: the (coarse) moduli space of genus 0 maps to G/P is a rational variety. The rationality argument depends essentially upon rationality results for quotients of SL2 representations proven by Katsylo and Bogomolov.

带扭场的叠理论与亏格2稳定映射模空间的消解

A theory of stacks with twisted fields and resolution of moduli of genus two stable maps

本文完整发展带扭场的叠（STF）理论，构造光滑代数叠实现亏格2到射影空间稳定映射模空间的奇点消解，还得到亏格1情形的新消解，是高亏格相关模空间消解研究的第二步。

Groups of automorphisms of Riemann surfaces and maps of genus $p+1$ where $p$ is prime

We classify compact Riemann surfaces of genus $g$, where $g-1$ is a prime $p$, which have a group of automorphisms of order $ρ(g-1)$ for some integer $ρ\ge 1$, and determine isogeny decompositions of the corresponding Jacobian varieties. This extends results of Belolipetzky and the second author for $ρ>6$, and of the first and third authors for $ρ=3, 4, 5$ and $6$. As a corollary we classify the orientably regular hypermaps (including maps) of genus $p+1$, together with the non-orientable regular hypermaps of characteristic $-p$, with automorphism group of order divisible by the prime $p$; thi…

通过非阿贝尔局部化研究旗流形的量子K-理论

Quantum K-theory of flag varieties via non-abelian localization

本文显式重构任意部分旗流形的亏格零K-理论Gromov-Witten不变量生成函数，推广环面局部化递归方法处理非孤立轨道，并证明扭曲函数性质及构造K-理论镜像。

The tautological rings of the moduli spaces of stable maps

We show that the rational cohomology of the genus zero stable map spaces to SL flag varieties is entirely tautological.

具有到固定椭圆曲线的每个度数映射的亏格2曲线

Curves of genus two with maps of every degree to a fixed elliptic curve

本文分类了所有亏格2曲线C和椭圆曲线E对，使得对每个整数n>1都存在从C到E的n次映射，并证明每条亏格2曲线都存在一个不超过59的整数n使得没有最小n次映射到椭圆曲线。

The genus zero Gromov-Witten invariants of [Sym^2 P^2]

We study the Abramovich--Vistoli moduli space of genus zero orbifold stable maps to [Sym^2 P^2], the stack symmetric square of P^2. This compactifies the moduli space of stable maps from hyperelliptic curves to P^2, and we show that all genus zero Gromov--Witten invariants are determined from trivial enumerative geometry of hyperelliptic curves. We also show how the genus zero Gromov--Witten invariants can be used to determine the number of hyperelliptic curves of degree d and genus g interpolating 3d + 1 generic points in P^2. Comparing our method to that of Graber for calculating the same nu…

The Geometry of Stable Quotients in Genus One

Stable quotient spaces provide an alternative to stable maps for compactifying spaces of maps. When the target is projective space and the domain curve has genus 1, these are smooth proper Deligne-Mumford stacks. In this paper we study the associated coarse moduli schemes. We show these schemes are projective, rationally connected and have Picard number 2. Then we give generators for the Picard group, compute the canonical divisor, and the cones of ample and effective divisors. In certain cases, we also give a closed formula for the Poincaré polynomial.

A Desingularization of the Main Component of the Moduli Space of Genus-One Stable Maps into $\Bbb{P}^n$

We construct a desingularization of the ``main component'' $\bar{\mathfrak M}_{1,k}^0(\Bbb{P}^n,d)$ of the moduli space $\bar{\mathfrak M}_{1,k}(\Bbb{P}^n,d)$ of genus-one stable maps into the complex projective space $\Bbb{P}^n$. As a bonus, we obtain desingularizations of certain natural sheaves over $\bar{\mathfrak M}_{1,k}^0(\Bbb{P}^n,d)$. Such desingularizations are useful for integrating natural cohomology classes on $\bar{\mathfrak M}_{1,k}^0(\Bbb{P}^n,d)$ using localization. In turn, these classes can be used to compute the genus-one Gromov-Witten invariants of complete intersections a…

到光滑对的扭曲映射模空间

Moduli spaces of twisted maps to smooth pairs

本文研究任意亏格下到光滑对的扭曲映射模空间，通过组合/热带信息分类其不可约分支，计算不同根参数下态射的次数并证明其多项式性，进而联系并推广了Tseng–You关于orbifold Gromov–Witten不变量的结果。

Gromov-Witten theory of root gerbes I: structure of genus $0$ moduli spaces

Let $X$ be a smooth complex projective algebraic variety. Given a line bundle $\mathcal{L}$ over $X$ and an integer $r>1$ one defines the stack $\sqrt[r]{\mathcal{L}/X}$ of $r$-th roots of $\mathcal{L}$. Motivated by Gromov-Witten theoretic questions, in this paper we analyze the structure of moduli stacks of genus $0$ twisted stable maps to $\sqrt[r]{\mathcal{L}/X}$. Our main results are explicit constructions of moduli stacks of genus $0$ twisted stable maps to $\sqrt[r]{\mathcal{L}/X}$ starting from moduli stack of genus $0$ stable maps to $X$. As a consequence, we prove an exact formula ex…

Abel maps for curves of compact type

Recently, the first Abel map for a stable curve of genus g>1 has been constructed. Fix an integer d>0 and let C be a stable curve of compact type of genus g>1. We construct two d-th Abel maps for C, having different targets, and we compare the fibers of the two maps. As an application, we get a characterization of hyperelliptic stable curves of compact type with two components via the 2-nd Abel map.

Universal relations on stable map spaces in genus zero

We introduce a factorization for the map between moduli spaces of stable maps which forgets one marked point. This leads to a study of universal relations in the cohomology of stable map spaces in genus zero.

On the canonical maps of nonsingular threefolds of general type

Let $S$ be a nonsingular minimal complex projective surface of general type and the canonical map of $S$ is generically finite. Beauville showed that the geometric genus of the image of the canonical map is vanishing or equals the geometric genus of $S$ and discussed the canonical degrees for these two cases. We generalize his results to nonsingular minimal complex projective threefolds.

Super quantum cohomology I: Super stable maps of genus zero with Neveu-Schwarz punctures

In this article we define stable supercurves and super stable maps of genus zero via labeled trees. We prove that the moduli space of stable supercurves and super stable maps of fixed tree type are quotient superorbifolds. To this end, we prove a slice theorem for the action of super Lie groups on Riemannian supermanifolds and discuss superorbifolds. Furthermore, we propose a Gromov topology on super stable maps such that the restriction to fixed tree type yields the quotient topology from the superorbifolds and the reduction is compact. This would, possibly, lead to the notions of super Gromo…

Wall-crossing in genus zero quasimap theory and mirror maps

For each positive rational number epsilon, the theory of epsilon-stable quasimaps to certain GIT quotients W//G developed in arXiv:1106.3724[math.AG] gives rise to a Cohomological Field Theory. Furthermore, there is an asymptotic theory corresponding to epsilon --> 0. For epsilon >1 one obtains the usual Gromov-Witten theory of W//G, while the other theories are new. However, they are all expected to contain the same information and in particular the numerical invariants should be related by wall-crossing formulas. In this paper we analyze the genus zero picture and find that the wall-crossing…

亏格0到光滑射影环簇的映射模空间的不可约分支

Irreducible components of moduli spaces of maps to smooth projective toric varieties in genus 0

本文通过阿贝尔锥的不可约分支研究，给出了亏格0到光滑射影环簇的映射模空间不可约分支的组合描述，并首次描述了非射影空间情形的可光滑轨迹。

The $r$-ELSV Formula via Localisation on the Moduli Space of Stable Maps with Divisible Ramification

The moduli space of stable maps with divisible ramification uses $r$-th roots of a canonical ramification section to parametrise stable maps whose ramification orders are divisible by a fixed integer $r$. In this article, a virtual fundamental class is constructed while letting domain curves have a positive genus; hence removing the restriction of the domain curves being genus zero. We apply the techniques of virtual localisation and obtain the $r$-ELSV formula as an intersection of the virtual class with a pullback via the branch morphism.

Reduced Genus-One Gromov-Witten Invariants

In a previous paper we described a natural closed subset of the moduli space of stable genus-one J-holomorphic maps into a symplectic manifold X. In this paper we generalize the definition of the main component to moduli spaces of perturbed, in a restricted way, J-holomorphic maps. This generalization implies that the main component, just like the entire moduli space, carries a virtual fundamental class and can be used to define symplectic invariants. These truly genus-one invariants constitute part of the standard genus-one Gromov-Witten invariants, which arise from the entire moduli space. T…

On the Genus-One Gromov-Witten Invariants of Complete Intersections

As shown in a previous paper, certain naturally arising cones of holomorphic vector bundle sections over the main component $\ov\M_{1,k}^0(¶,d)$ of the moduli space of stable genus-one holomorphic maps into $¶$ have a well-defined euler class. In this paper, we extend this result to moduli spaces of perturbed, in a restricted way, $J$-holomorphic maps. We show that euler classes of such cones relate the reduced genus-one Gromov-Witten invariants of complete intersections to the corresponding GW-invariants of the ambient projective space. As a consequence, the standard genus-one GW-invariants o…

零亏格超稳定映射的模空间上的环面作用

Torus Actions on Moduli Spaces of Super Stable Maps of Genus Zero

本研究在零亏格固定树型的超J-全纯曲线等三类模空间上构造了光滑$\u2102^*$-作用并保证约化空间环面不变，还给出固定点法丛的显式描述，明确了构造的核心步骤。