Dan Abramovich, Angelo Vistoli
We define two equivalent notions of twisted stable map from a curve to a Deligne-Mumford stack with projective moduli space, and we prove that twisted stable maps of fixed degree form a complete Deligne-Mumford stack with projective moduli space.
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In this paper, we compare the compactified Torelli morphism (as defined by V. Alexeev) and the tropical Torelli map (as defined by the author in a joint work with S. Brannetti and M. Melo, and furthered studied by M. Chan). Our aim is twofold: on one hand, we will review the construction and main properties of the above mentioned two Torelli maps, focusing in particular on the description of their fibers achieved by the author in joint works with L. Caporaso; on the other hand, we will clarify the relationship between the two Torelli maps via the introduction of the reduction maps and the trop…
Compactifying the rank two Hitchin system via spectral data on semistable curves
本文通过二次多重尺度微分空间，利用带基点稳定曲线的紧化雅可比簇，提供了秩二 Hitchin 系统的紧化，并将经典的谱对应扩展到了紧化的 Hitchin 系统。
Compactifying real analytic functions and resulting Reeb spaces
自然地对连续映射进行紧致化，主要考虑实解析函数，通过里布空间研究其拓扑与组合性质，此前已获相关成果，现给出新讨论与例子。
We study algebraic (Artin) stacks over $\bar{\mathcal M}_g$ giving a functorial way of compactifying the relative degree $d$ Picard variety for families of stable curves. We also describe for every $d$ the locus of genus $g$ stable curves over which we get Deligne-Mumford stacks strongly representable over $\bar{\mathcal M}_g$.
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.
We consider the Quot scheme, R_{d}, compactifying the space of degree d maps from the projective line to the Grassmannian of lines. We give an algorithm for computing the degree of R_{d} under a "generalized Plücker embedding", this is a certain Gromov-Witten invariant. The approach is to apply the Atiyah-Bott localization formula for the natural C^{*}-action on R_{d}. These numbers can be obtained directly from Vafa-Intriligator formula.
In this article we announce some results on compactifying moduli spaces of rank-2 vector bundles on surfaces by spaces of vector bundles on trees of surfaces. This is thought as an algebraic counterpart of the so called bubbling of vector bundles and connections in differential geometry. The new moduli spaces are algebraic spaces arising as quotients by group actions according to a result of Kollár. As an example the compactification of the space of stable rank-2 vector bundles with Chern classes $c_1=0, c_1=2$ on the projective plane is studied in more detail. Proofs are only indicated and wi…
For any moduli space of stable representations of quivers, certain smooth varieties, compactifying projective space fibrations over the moduli space, are constructed. The boundary of this compactification is analyzed. Explicit formulas for the Betti numbers of the smooth models are derived. In the case of moduli of simple representations, explicit cell decompositions of the smooth models are constructed.
The evaluation stack for minimal logarithmic stable maps is constructed, parameterizing families of standard log points in the target log scheme. This construction provides the ingredients necessary to define appropriate evaluation maps for minimal log stable maps and establish the logarithmic Gromov-Witten theory of a log-smooth Deligne- Faltings log scheme.
We analyze the relationship between two compactifications of the moduli space of maps from curves to a Grassmannian: the Kontsevich moduli space of stable maps and the Marian--Oprea--Pandharipande moduli space of stable quotients. We construct a moduli space which dominates both the moduli space of stable maps to a Grassmannian and the moduli space of stable quotients, and equip our moduli space with a virtual fundamental class. We relate the virtual fundamental classes of all three moduli spaces using the virtual push-forward formula. This gives a new proof of a theorem of Marian-Oprea-Pandha…
In this note we give a definition of stable maps into the classifying stack $\BGL_r$ of the general linear group. To support our belief that the definition is the correct one, we show that there are natural boundary morphisms between the moduli groupoids parameterizing stable maps to $\BGL_r$.
We prove a virtual localization formula for Bumsig Kim's space of logarithmic stable maps. The formula is closely related and can in fact recover the relative virtual localization formula of Graber and Vakil.
It is known that the main component of the Kontsevich's moduli space of elliptic stable maps is singular. There are two different desingularizations. One is Vakil-Zinger's desingularization and the other is the moduli space of logarithmic stable maps. When the degree is less then or equal to 3 and the target is $\mathbb{P}^{n}$, we show that the moduli space of logarithmic stable maps can be obtained by blowing up Vakil-Zinger's desingularization.
By using Oprea's Bialynicki-Birula decomposition for the stack of genus zero stable maps to flag manifolds. We calculate the Poincaré polynomial of the moduli space in degree one and degree two.
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.
Twisted stable maps with colliding points
本文研究了允许标记点重合的带点曲线到驯顺 Deligne-Mumford 栈的稳定映射模空间，推广了 Abramovich-Vistoli 扭曲稳定映射理论及相关带权标记点曲线到射影簇的稳定映射研究。
Using the equivariant virtual cycle of the moduli space of stable maps to [C/Z_r], or equivalently, the vanishing of high-degree Chern classes of a certain vector bundle over the moduli space of stable maps to BZ_r, we derive relations in the Chow ring of Mbar_{g,n}(BZ_r,0). These push forward to yield tautological relations on Mbar_{g,n}.
Birational maps of moduli spaces of stable sheaves on a K3 surface induced by Fourier-Mukai transforms
该研究探讨由Fourier-Mukai变换诱导的皮卡数为1的K3曲面上稳定层模空间的双有理映射问题。
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.
We present a localization proof of the fact that the cohomology of the moduli spaces of genus zero stable maps to projective spaces is entirely tautological. In addition, we obtain a description of a Bialynicki-Birula stratification in the context of Gathmann's moduli spaces of maps with prescribed contact order to a fixed hyperplane.
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.
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…
This paper provides a GIT construction of the Moduli Space of Stable Maps as a GIT quotient of the Graph Space by SL(2,C). As a corollary, we get a birational map from the 0-pointed Moduli Space to a projective variety.
We show that the rational cohomology of the genus zero stable map spaces to SL flag varieties is entirely tautological.
In this paper we compute the multiple cover Gromov-Witten integrals (analog of the Aspinwall-Morrison formula) for the unramified compactification of the moduli space of stable maps to an embedded $\PO$ in a Calabi-Yau threefold $X$ with the normal bundle $\mathcal{O}_{\PO}(-1) \bigoplus \mathcal{O}_{\PO}(-1)$.
We introduce a new compactification of the space of relative stable maps. This new method uses logarithmic geoemtry in the sense of Kato-Fontaine-Illusie rather than the expanded degeneration. The underlying structure of our log stable maps is stable in the usual sense.
Let X be a smooth projective Deligne-Mumford stack over an algebraically closed field k of characteristic 0. In this paper we construct the moduli stack of very twisted stable maps, extending the notion of twisted stable maps by Abramovich and Vistoli to allow for generic stabilizers on the source curves. We also consider the Gromov-Witten theory given by this construction.
We determine generators for the codimension 1 Chow group of the moduli spaces of genus zero stable maps to flag varieties G/P. In the case of SL flags, we find all relations between our generators, showing that they essentially come from $\bar M_{0,n}$. In addition, we analyze the codimension 2 classes on the moduli spaces of stable maps to Grassmannians and prove a new codimension 2 relation. This will lead to a partial reconstruction theorem for the Grassmannian of 2 planes.
We introduce and compute the class of a number of effective divisors on the moduli space of stable maps $\bar M_{0,0}(P^{r},d)$, which, for small d, provide a good understanding of the extremal rays and the stable base locus decomposition for the effective cone. We also discuss various birational models that arise in Mori's program, including the Hilbert scheme, the Chow variety, the space of $k$-stable maps, the space of branchcurves and the space of semi-stable sheaves.
In a previous paper, the author and David Swinarski constructed the moduli spaces of stable maps, \bar M_g,n(P^r,d), via geometric invariant theory (GIT). That paper required the base field to be the complex numbers, a restriction which this paper removes: here the coarse moduli spaces of stable maps are constructed via GIT over a more general base.
We present a new compactification $M(d,n)$ of the moduli space of self-maps of $\mathbb{CP}^1$ of degree $d$ with $n$ markings. It is constructed via GIT from the stable maps moduli space $\ ar M_{0,n}(\mathbb{CP}^1 \times \mathbb{CP}^1, (1,d))$. We show that it is the coarse moduli space of a smooth Deligne-Mumford stack and we compute its rational Picard group. Using the recursive boundary structure inherited from the stable maps space, we give an explicit algorithm for computing top-intersection numbers of divisors on $M(d,n)$. We also study the $m$-fold iteration map $M(d,n) \dashrightarro…
We construct moduli spaces of rational covers of an arbitrary smooth tropical curve in R^r as tropical varieties. They are contained in the balanced fan parametrizing tropical stable maps of the appropriate degree to R^r. The weights of the top-dimensional polyhedra are given in terms of certain lattice indices and local Hurwitz numbers.
Smoothability of relative stable maps to stacky curves
本文利用log几何研究了堆叠曲线上的稳定映射光滑性，并应用于平滑半双有理纤维面。
We classify the Deligne-Mumford stacks M compactifying the moduli space of smooth $n$-pointed curves of genus one under the condition that the points of M represent Gorenstein curves with distinct markings. This classification uncovers new moduli spaces $\bar{\mathcal{M}}_{1,n}(Q)$, which we may think of coming from an enrichment of the notion of level used to define Smyth's $m$-stable spaces. Finally, we construct a cube complex of Artin stacks interpolating between the $\bar{\mathcal{M}}_{1,n}(Q)$'s, a multidimensional analogue of the wall-and-chamber structure seen in the log minimal model …
We define tautological relations for the moduli space of stable maps to a target variety. Using the double ramification cycle formula for target varieties of Janda-Pandharipande-Pixton-Zvonkine, we construct nontrivial tautological relations parallel to Pixton's double ramification cycle relations for the moduli of curves. Examples and applications are discussed.
Let $X$ be a projective normal toric variety and $T_0$ a rank one subtorus of the defining torus of $X$. We show that the normalization of the Chow quotient $X//T_0$, in the sense of Kapranov-Sturmfels-Zelevinsky, coarsely represents the moduli space of stable log maps to $X$ with discrete data given by $T_0\subset X$.
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.
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.
Relations for quadratic Hodge integrals via stable maps
该研究遵循Faber-Pandharipande方法，利用到$ \mathbb{P}^1 $的稳定映射模空间虚局部化公式计算霍奇积分关系，证明其某些生成函数为多项式，为二次霍奇积分研究提供新结果。
We construct a moduli space of stable maps with fields associated to a triple $(X,E,s)$ of a projective variety (or a DM stack with projective moduli space) a vector bundle and a section. We show the class coincides up to a sign with the virtual fundamental class on the moduli space of stable maps to $Z=Z(s)\subset X$. We show that this gives a generalization of the Quantum Lefschetz hyperplane principle. This generalizes similar comparison results of Chang--Li, Kim--Oh and Chang--Li and presents a different approach from Chen--Janda--Webb.
We use the theory of twisted stable maps to Deligne-Mumford stacks to construct compactifications of the moduli space of pairs $(X \to C, S + F)$ where $X \to C$ is a fibered surface, $S$ is a sum of sections, $F$ is a sum of marked fibers, and $(X, S + F)$ is a stable pair in the sense of the minimal model program. This generalizes the work of Abramovich-Vistoli, who compactified the moduli space of fibered surfaces without marked fibers. Furthermore, we compare our compactification to Alexeev's space of stable maps and the KSBA compactification of stable pairs. As an application, we describe…
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.
We generalize the results of Chang-Li, Kim-Oh and Chang-Li on the moduli of $p$-fields to the setting of (quasi-)maps to complete intersections in arbitrary smooth Deligne-Mumford stacks with projective coarse moduli. In particular, we show that the virtual cycle of stable (quasi-)maps to a complete intersection can be recovered by the cosection localized virtual cycle of the moduli of $p$-fields of the ambient space.
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 $\mathbf{M}_0(P,3)$ …
We study the Berkovich analytification of the space of genus $0$ logarithmic stable maps to a toric variety $X$ and present applications to both algebraic and tropical geometry. On algebraic side, insights from tropical geometry give two new geometric descriptions of this space of maps -- (1) as an explicit toroidal modification of $\overline M_{0,n}\times X$ and (2) as a tropical compactification in a toric variety. On the combinatorial side, we prove that the tropicalization of the space of genus $0$ logarithmic stable maps coincides with the space of tropical stable maps, giving a large new…
We compare the Kontsevich moduli space of genus 0 stable maps to projective space with the quasi-map space when $d=3$. More precisely, we prove that when $d=3$, the obvious birational map from the quasi-map space to the moduli space of stable maps is the composition of three blow-ups followed by two blow-downs. Furthermore, we identify the blow-up/down centers explicitly in terms of the moduli spaces for lower degrees. Using this, we calculate the Betti numbers, the integral Picard group, and the rational cohomology ring. The degree two case is worked out as a warm-up.
According to the Bloch-Beilinson conjectures, an automorphism of a K3 surface X that acts as the identity on the transcendental lattice should act trivially on CH^2(X). We discuss this conjecture for symplectic involutions and prove it in one third of all cases. The main point is to use special elliptic K3 surfaces and stable maps to produce covering families of elliptic curves on the generic K3 surface that are invariant under the involution.
We use stable maps, and their stable lifts to the Semple bundle variety of second-order curvilinear data, to calculate certain characteristic numbers for rational plane curves. These characteristic numbers involve first-order (tangency) and second-order (inflectional) conditions. Although they may be virtual, they may be used as inputs in an enumeratively significant formula for the number of rational curves having a triple contact with a specified plane curve and passing through 3d-3 general points.
Stable maps to quotient stacks with a properly stable point
本文研究具有射影好模空间的商栈上曲线映射的模空间紧化，提出扩展加权吹胀的双有理操作，将拟映射理论紧化扩展至多类映射与纤维化模空间，并在附录中模块化证明Hassett猜想。
We prove that the moduli space of stable maps of degree 2 to the moduli space of rank 2 stable bundles of fixed determinant O(-x) over a smooth projective curve of genus g>2 has two irre- ducible components which intersect transversely. One of them is Kir- wan's partial desingularization of the moduli space of rank 2 semistable bundles with determinant isomorphic to O(y-x) for some y in X. The other component is the partial desingularization of PHom(sl(2)^*;W)//PGL(2) for a vector bundle W of rank g over the Jacobian of X. We also show that the Hilbert scheme H, the Chow scheme C of conics in …
We consider two cycles on the moduli space of compact type curves and prove that they coincide. The first is defined by pushing forward the virtual fundamental classes of spaces of relative stable maps to an unparameterized rational curve, while the second is obtained as the intersection of the Abel section of the universal Jacobian with the zero section. Our comparison extends results of Cavalieri-Marcus-Wise where the same identity was proved over on the locus of rational tails curves.
We show that algebraic equivalence of images of stable maps of curves lifts to deformation equivalence of the stable maps. The main applications concern $A_1(X)$, the group of 1-cycles modulo algebraic equivalence, for smooth, separably rationally connected varieties. If $K/k$ is an algebraic extension, then the kernel of $A_1(X_k)\to A_1(X_K)$ is at most ${\mathbb Z}/2{\mathbb Z}$. If $k$ is finite, then the image equals the subgroup of Galois invariant cycles. This paper replaces Sections~2--3 of 2211.15915.v.1 and Sections~2--3 of 2211.15911.v.1. The other Sections are retained in the revis…
We construct a natural branch divisor for equidimensional projective morphisms where the domain has lci singularities and the target is nonsingular. The method involves generalizing a divisor contruction of Mumford from sheaves to complexes. The construction is valid in flat families. The generalized branch divisor of a stable map to a nonsingular curve X yields a canonical morphism from the space of stable maps to a symmetric product of X. This branch morphism (together with virtual localization) is used to compute the Hurwitz numbers of covers of P^1 for all genera and degrees in terms of Ho…
We prove that the moduli space of stable logarithmic maps with fixed numerical invariants, from logarithmic curves to a fixed projective target logarithmic scheme with fine and saturated logarithmic structure, is a proper algebraic stack. This was previously known only with further restrictions on the logarithmic structure of the target.
Choosing a normal crossings anticanonical divisor of $\mathbb{P}^2$ leads to four log Calabi-Yau surfaces, three of which are Looijenga pairs. In this survey article, I describe how to count rational curves in these that intersect each component of the divisor in one point of maximal tangency.
We make an observation which enables one to deduce the existence of an algebraic stack of log maps for all generalized Deligne--Faltings log structures (in particular simple normal crossings divisor) from the simplest case with log structures given by a Cartier divisor (essentially the smooth divisor case).
We construct Abel maps for a stable curve $X$. Namely, for each one-parameter deformation of $X$ with regular total space, and every integer $d>0$, we construct by specialization a map $α^d_X$ from the smooth locus of $X^d$ to the moduli scheme of balanced line bundles on semistable curves over $X$. For $d=1$, we show that $α^1_X$ naturally extends over $X$, and does not depend on the choice of the deformation; we give a precise description of when it is injective.
A component of the moduli space M_g(Y,b) of stable maps from genus g curves to a variety Y is said to be regular if it is generically smooth and of the expected dimension provided by deformation theory. In this note we prove existence of regular components of M_g(Y,b) when Y is a product of projective spaces. This result can also be seen as a general position theorem for Brill-Noether loci in M_g.
Stable maps, multiplicities, and compactified Jacobians
研究复杂积分曲线的稳定映射、变形空间和紧化雅可比簇之间的数值关系，修正Fantechi-Göttsche-van Straten关于变形空间δ-常数层在[C]处乘积性的陈述，并给出原主张成立的充要条件。
We give an introduction to moduli stacks of gauged maps satisfying a stability conditition introduced by Mundet and Schmitt, and the associated integrals giving rise to gauged Gromov-Witten invariants. We survey various applications to cohomological and K-theoretic Gromov-Witten invariants.
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