A. Zinger
We give a formula computing the number of one-nodal rational curves that pass through an appropriate collection of constraints in a complex projective space. We combine the methods and results from three different papers.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Enumeration of non-nodal real plane rational curves
该研究探讨计数带非结点奇点的实有理平面曲线的不变量，证明唯一的此类不变量对应实有理三尖点四次曲线，且任意四对复共轭点的一般构型均存在一对这类曲线。
We give a practical formula for counting irreducible nodal genus-three plane curves that a fixed generic complex structure on the normalization. As an intermediate step, we enumerate rational plane curves that have a $(3,4)$-cusp.
We continue the development of methods for enumerating nodal curves on smooth complex surfaces, stressing the range of validity. We illustrate the new methods in three important examples. First, for up to eight nodes, we confirm Göttsche's conjecture about plane curves of low degree. Second, we justify Vainsencher's enumeration of irreducible six-nodal plane curves on a general quintic threefold in four-space. Third, we supplement Bryan and Leung's enumeration of nodal curves in a given homology class on an Abelian surface of Picard number one.
Tropical enumeration of curves in blowups of the projective plane
提出一种利用tropical曲线和相对Gromov-Witten不变量递归计算射影平面所有爆破的Gromov-Witten不变量的方法，该递归公式不同于已有在零亏格和无爆破情形下的公式。
The family of complex projective surfaces in projective three space of degree $d$ having precisely $δ$ nodes as their only singularities has codimension $δ$ in the linear system of surfaces of degree $d$ for sufficiently large $d$ and is of degree $N_{δ,complex}(d)=(4(d-1)^3)^δ/δ!+O(d^{3δ-3})$. In particular, this number is polynomial in $d$. By means of tropical geometry, we explicitly describe $(4d^3)^δ/δ!+O(d^{3δ-1})$ surfaces passing through a suitable generic configuration of $n=\binom{d+3}{3}-δ-1$ points in projective three space. These surfaces are close to tropical limits which we char…
We provide an inductive algorithm computing Gromov-Witten invariants in all genera with arbitrary insertions of all smooth complete intersections in projective space. We also prove that all Gromov-Witten classes of all smooth complete intersections in projective space belong to the tautological ring of the moduli space of stable curves. The main idea is to show that invariants with insertions of primitive cohomology classes are controlled by their monodromy and by invariants defined without primitive insertions but with imposed nodes in the domain curve. To compute these nodal Gromov-Witten in…
Enumeration of Elliptic Curves via Elliptic Gromov-Witten Invariants of Four Dimensional Projective Fano Hypersurfaces
本文通过四维射影Fano超曲面的椭圆GW不变量，提出一个猜想，阐明椭圆曲线数量与GW不变量之间的关系。
Let $U^{'s}_L(n,d)$ be the moduli space of stable vector bundles of rank $n$ with determinant $L$ where $L$ is a fixed line bundle of degree $d$ over a nodal curve $Y$. We prove that the projective Poincare bundle on $Y \times U^{'s}_L(n,d)$ and the projective Picard bundle on $U^{'s}_L(n,d)$ are stable for suitable polarisation. For a nonsingular point $x \in Y$, we show that the restriction of the projective Poincare bundle to $x \times U^{'s}_L(n,d)$ is stable for any polarisation. We prove that for arithmetic genus $g\ge 3$ and for $g=n=2, d$ odd, the Picard group of the moduli space $U'_L…
In the long paper "Family Blowup formula, Admissible Graphs and the Enumeration of Singular Curves (I)" (appearing in JDG), the author solved the enumeration problem of nodal (or general singular) curve counting on algebraic surfaces by using techniques from differential topology/symplectic geometry (including family Seiberg-Witten theory) and some ideas derived from Taubes' "SW=Gr". In the current paper, we offer an algebraic proof of the "universality theorem", showing that the counting of nodal curves for 5$δ$-1 very ample complete linear systems are controled by universal polynomials of th…
In this Part II, D(10.2), of D(10), we take D(10.1) (arXiv:1302.2054 [math.AG]) as the foundation to define the notion of $Z$-semistable morphisms from general Azumaya nodal curves, of genus $\ge 2$, with a fundamental module to a projective Calabi-Yau 3-fold and show that the moduli stack of such $Z$-semistable morphisms of a fixed type is compact. This gives us a counter moduli stack to D-strings as the moduli stack of stable maps in Gromov-Witten theory to the fundamental string. It serves and prepares for us the basis toward a new invariant of Calabi-Yau 3-fold that captures soft-D-string …
We give the full answer to the question: on which curves the category of coherent sheaves $\Coh_{X}$ is tame. The answer is: these are just the curves from the list of Drozd-Greuel. Moreover, in this case the derived category $D^{-}(\Coh_{X})$ is also tame. We give an explicit description of the objects of this category as well as of the categories $D^{b}(\Coh_{X})$, $\Coh_{X}$. Among the coherent sheaves we describe the vector bundles, torsion-free sheaves, mixed sheaves and skyscraper sheaves.
In a suitable regime of superstring theory, D-branes in a Calabi-Yau space and their most fundamental behaviors can be nicely described mathematically through morphisms from Azumaya spaces with a fundamental module to that Calabi-Yau space. In the earlier work [L-L-S-Y] (D(2): arXiv:0809.2121 [math.AG], with Si Li and Ruifang Song) from the project, we explored this notion for the case of D1-branes (i.e. D-strings) and laid down some basic ingredients toward understanding the notion of D-string world-sheet instantons in this context. In this continuation, D(10), of D(2), we move on to construc…
We establish a homology relation for the Deligne-Mumford moduli spaces of real curves which lifts to a WDVV-type relation for real Gromov-Witten invariants of real symplectic manifolds; we also obtain a vanishing theorem for these invariants. For many real symplectic manifolds, these results reduce all genus 0 real invariants with conjugate pairs of constraints to genus 0 invariants with a single conjugate pair of constraints. In particular, we give a complete recursion for counts of real rational curves in odd-dimensional projective spaces with conjugate pairs of constraints and specify all c…
Let $Y_{1},\dots,Y_{l}$ be smooth irreducible projective curves and let $Y$ be its disjoint union. Given a semisimple reductive algebraic group $G$ and a faithful representation $ρ:G\hookrightarrow \textrm{SL}(V)$ we construct a projective moduli space of $(\underlineκ,δ)$-(semi)stable singular principal $G$-bundles with generalized parabolic structure of type $\underline{e}$. In case $Y$ is the normalization of a connected and reducible projective nodal curve $X$, there is a closed subscheme coarsely representing the subfunctor corresponding to descending bundles. We prove that the descent op…
In this note we compute some enumerative invariants of real and complex projective spaces by means of some enriched graphs called floor diagrams.
In this paper we partially address two issues: - The first is a rigidity property for pairs (S,C) consisting of a general projective K3 surface S, and a curve C obtained as the normalization of a nodal, hyperplane section of S. We prove that a non-trivial deformation of such a pair (S,C) induces a non-trivial deformation of C; - The second question concerns the Wahl map of curves C as above. We prove that the Wahl map of the normalization of a nodal curve contained in a general projective K3 surface is non-surjective. In both cases, we impose upper bounds on the number of nodes of the hyperpla…
We prove the factoriality of a nodal hypersurface in $\mathbb{P}^{4}$ of degree $d$ that has at most $2(d-1)^{2}/3$ singular points, and factoriality of a double cover of $\mathbb{P}^{3}$ branched over a nodal surface of degree $2r$ having less than $(2r-1)r$ singular points.
For each $1\leq n\leq6$ we present formulas for the number of $n-$nodal curves in an $n-$dimensional linear system on a smooth, projective surface. This yields in particular the numbers of rational curves in the system of hyperplane sections of a generic $K3-$surface imbedded in \p{n} by a complete system of curves of genus $n$ as well as the number {\bf17,601,000} of rational ({\em singular}) plane quintic curves in a generic quintic threefold.
We use floor decompositions of tropical curves to prove that any enumerative problem concerning conics passing through projective-linear subspaces in $\RP^n$ is maximal. That is, there exist generic configurations of real linear spaces such that all complex conics passing through these constraints are actually real.
Conical resolutions and cohomology of the moduli spaces of nodal hypersurfaces
本文利用锥形消解及其推广形式，并结合紧 GIT 商的显式描述，计算节点三次、四次平面曲线及三次曲面方程空间与模空间的有理上同调。
Let $V_{n,d}$ be the variety of equations for hypersurfaces of degree $d$ in $\mathbb{P}^n(\mathbb{C})$ with singularities not worse than simple nodes. We prove that the orbit map $G'=SL_{n+1}(\mathbb{C}) \to V_{n,d}$, $g\mapsto g\cdot s_0$, $s_0\in V_{n,d}$ is surjective on the rational cohomology if $n>1$, $d\geq 3$, and $(n,d)\neq (2,3)$. As a result, the Leray-Serre spectral sequence of the map from $V_{n,d}$ to the homotopy quotient $(V_{n,d})_{hG'}$ degenerates at $E_2$, and so does the Leray spectral sequence of the quotient map $V_{n,d}\to V_{n,d}/G'$ provided the geometric quotient $V…
We give a bound on the minimal number of singularities of a nodal projective complete intersection threefold which contains a smooth complete intersection surface that is not a Cartier divisor.
The main purpose of this paper is to introduce a new approach to study families of nodal curves on projective threefolds. Precisely, given $X$ a smooth projective threefold, $\E$ a rank-two vector bundle on $X$, $L$ a very ample line bundle on $X$ and $k \geq 0$, $δ>0 $ integers and denoted by $V= {\V}_δ ({\E} \otimes L^{\otimes k})$ the subscheme of ${\Pp}(H^0({\E} \otimes L^{\otimes k}))$ parametrizing global sections of ${\E} \otimes L^{\otimes k}$ whose zero-loci are irreducible and $δ$-nodal curves on $X$, we present a new cohomological description of the tangent space $T_{[s]}({\V}_δ ({\…
In this paper, we study the enumeration of virtual numbers of immersed nodal curves along certain Calabi-Yau K3 fibrations. By using the concept of cosmic strings, we verify the modularity conjeture of the generating function of immersed nodal curves from String theory (Harvey-Moore). Based on the theory defining virtual numbers of nodal curves on algebraic surface (applying to a pencil of lattice polarized K3), a mathematical definition of virtual numbers of immersed rational curves (so-called Gopakumar-Vafa numbers) for Calabi-Yau K3 fibrations is given and it is shown to match up with strin…
We generalize Bertram's work on rank two vector bundles to an irreducible projective nodal curve C. We use extensions of a line bundle L by O_C and the associated `forgetful' map to study a compactification of the moduli space of semi-stable vector bundles of rank 2 and determinant L on C.
We find a geometrical method of analysing the singularities of a plane nodal curve. The main results will be used in a forthcoming paper on geometric Plucker formulas for such curves. Plane nodal curves, that is plane curves having at most nodes as singularities, form an important class of curves, as any projective algebraic curve is birational to a plane nodal curve.
Welschinger's invariant bounds from below the number of real rational curves through a given generic collection of real points in the real projective plane. We estimate this invariant using Mikhalkin's approach which deals with a corresponding count of tropical curves. In particular, our estimate implies that, for any positive integer $d$, there exists a real rational curve of degree $d$ through any collection of $3d-1$ real points in the projective plane, and, moreover, asymptotically in the logarithmic scale at least one third of the complex plane rational curves through a generic point coll…
Enumeration of rational curves in a moving family of $\mathbb{P}^2$
本文针对 $\mathbb{P}^3$ 中位于移动 $\mathbb{P}^2$ 内并穿过指定点、线的有理曲线，给出满足 $r+2s=3d+2$ 的数目递推公式，并验证其支持 Laarakker 结点曲线多项式的枚举性猜想。
A very general surface of degree at least four in projective space of dimension three contains no curves other than intersections with surfaces. We find a formula for the degree of the locus of surfaces of degree at least five which contain some elliptic quartic curve. We also compute the degree of the locus of quartic surfaces containing an elliptic quartic curve, a case not covered by that formula.
Let $X$ be a smooth projective surface and $L\in \mathrm{Pic}(X)$. We prove that if $L$ is $(2k-1)$-spanned, then the set $\tilde{V}_k(L)$ of all nodal and irreducible $D\in |L|$ with exactly $k$ nodes is irreducible. The set $\tilde{V}_k(L)$ is an open subset of a Severi variety of $|L|$, the full Severi variety parametrizing all integral $D\in |L|$ with geometric genus $g(L)-k$.
Mehta and Seshadri have proved that the set of equivalence classes of irreducible unitary representations of the fundamental group of a punctured compact Riemann surface, can be identified with equivalence classes of stable parabolic bundles of parabolic degree zero on the compact Riemann surface. In this note, we discuss the Mehta-Seshadri correspondence over an irreducible projective curve with at most nodes as singularities.
In this paper we focus on the problem of computing the number of moduli of the so called Severi varieties (denoted by V(|D|, δ)), which parametrize universal families of irreducible, δ-nodal curves in a complete linear system |D|, on a smooth projective surface S of general type. We determine geometrical and numerical conditions on D and numerical conditions on δensuring that such number coincides with dim(V(|D|, δ). As related facts, we also determines some sharp results concerning the geometry of some Severi varieties.
Homological mirror symmetry for nodal stacky curves
本研究通过将B模型视为栈式射影线的环、推广已有方法使曲线不可约分支具非平凡一般稳定子群，证明了A模型为可逆曲线奇点米尔诺纤维有限商的同调镜像对称，验证了相关猜想，所得中间结果也具独立价值。
We study the relative Hilbert scheme of a family of nodal (or smooth) curves, over a base of arbitrary dimension, via its (birational) cycle map, going to the relative symmetric product. We show the cycle map is the blowing up of the discriminant locus, which consists of cycles with multiple points. We determine the relevant cotangent sheaves and complexes. We determine the structure of certain projective bundles called node scrolls, which play an important role in the geometry of Hilbert schemes.
Enumeration of rational contact curves via torus actions
本文利用环面作用和Bott公式证明了任意切触复射影空间中有理切触曲线的Gromov-Witten数具有枚举性，并计算了低次情形下的精确值。
Welschinger invariants of the real projective plane can be computed via the enumeration of enriched graphs, called marked floor diagrams. By a purely combinatorial study of these objects, we prove a Caporaso-Harris type formula which allows one to compute Welschinger invariants for configurations of points with any number of complex conjugated points.
Let X be a smooth projective surface. Here we study the postulation of a general union Z of fat points of X, when most of the connected components of Z have multiplicity 2. This problem is related to the existence of "good" families of curves on X, with prescribed singularities, most of them being nodes, and to the cohomology of suitable line bundles on blowing ups of X. More precise statements are obtained in the case X=P^2.
In this article we introduce a new class of non-commutative projective curves and show that in certain cases the derived category of coherent sheaves on them has a tilting complex. In particular, we prove that the right bounded derived category of coherent sheaves on a reduced rational projective curve with only nodes and cusps as singularities, can be fully faithfully embedded into the right bounded derived category of the finite dimensional representations of a certain finite dimensional algebra of global dimension two. As an application of our approach we show the dimension of the bounded d…
One of the earliest results in enumerative combinatorial geometry is the following theorem of de Bruijn and Erdős: Every set of points $E$ in a projective plane determines at least $|E|$ lines, unless all the points are contained in a line. Motzkin and others extended the result to higher dimensions, who showed that every set of points $E$ in a projective space determines at least $|E|$ hyperplanes, unless all the points are contained in a hyperplane. Let $E$ be a spanning subset of a $d$-dimensional vector space. We show that, in the partially ordered set of subspaces spanned by subsets of $E…
This paper contains an attempt to formulate rigorously and to check predictions in enumerative geometry of curves following from Mirror Symmetry. The main tool is a new notion of stable map. We give an outline of a contsruction of Gromov-Witten invariants for all algebraic projective or closed symplectic manifolds. Mirror Symmetry in the basic example of rational curves on a quintic 3-folds is reduced to certain complicated but explicit identity. The strategy of computations can be described as follows: 1) we reduce counting problems to questions concerning Chern classes on spaces of curves on…
We generalize a formula due to Macdonald that relates the singular Betti numbers of $X^{n}/G$ to those of $X$, where $X$ is a compact manifold and $G$ is any subgroup of the symmetric group $S_{n}$ acting on $X^{n}$ by permuting coordinates. Our result is completely axiomatic: in a general setting, given an endomorphism on the cohomology $H^{\bullet}(X)$, it explains how we can explicitly relate the Lefschetz series of the induced endomorphism on $H^{\bullet}(X^{n})^{G}$ to that of the given endomorphism on $H^{\bullet}(X)$ in the presence of the Künneth formula with respect to a cup product. …
An Equisingular Specialisation of the Compactified Jacobian and its applications
该文证明带 $k$ 个节点的不可约节点曲线的紧致雅可比簇的贝蒂数与混合霍奇数同于正规化雅可比簇与 $k$ 个单节点有理曲线之积，通过构造拓扑局部平凡族实现。
On a fibre bundle version of the Caporaso-Harris formula
本文推广Caporaso-Harris公式，提出纤维丛版本，并解决P^3中平面节点三次曲线特征数计数问题。
We develop a version of Mikhalkin's lattice path algorithm for projective hypersurfaces of arbitrary degree and dimension, which enumerates singular tropical hypersurfaces passing through appropriate configuration of points. By proving a correspondence theorem combined with the lattice path algorithm, we construct a $δ$ dimensional linear space of degree $ d $ real hypersurfaces containing $\frac{1}{δ!} (γ_nd^n)^δ+ O(d^{nδ-1})$ hypersurfaces with $ δ$ real nodes, where $ γ_n $ are positive and given by a recursive formula. This is asymptotically comparable to the number $ \frac{1}{δ!} \left( (…
We prove a correspondence theorem for singular tropical surfaces in real three space, which recovers singular algebraic surfaces in an appropriate toric three-fold that tropicalize to a given singular tropical surface. Furthermore, we develop a three-dimensional version of Mikhalkin's lattice path algorithm that enumerates singular tropical surfaces passing through an appropriate configuration of points in real three space. As application we show that there are pencils of real surfaces of degree $d$ in projective three space containing at least $(3/2)d^3+O(d^2)$ singular surfaces, which is asy…
In this paper we obtain an explicit formula for the number of curves in two dimensional complex projective space, of degree d, passing through d(d+3)/2-(k+1) generic points and having one node and one codimension k singularity, where k is at most 6. Our main tool is a classical fact from differential topology: the number of zeros of a generic smooth section of a vector bundle V over M, counted with a sign, is the Euler class of V evaluated on the fundamental class of M.
Weil's Theorem for Logarithmic Connections on Irreducible Nodal Curves
本研究将André Weil经典定理推广至不可约射影结点曲线，证明其上不可分解向量丛或无挠凝聚层存在关于对偶化层的全纯对数联络等价于次数为0，还构造了有理结点三次曲线上的显式单参数平坦联络族。
Classification theory and the study of projective varieties which are covered by rational curves of minimal degrees naturally leads to the study of families of singular rational curves. Since families of arbitrarily singular curves are hard to handle, it has been shown in a previous paper that there exists a partial resolution of singularities which transforms a bundle of possibly badly singular curves into a bundle of nodal and cuspidal plane cubics. In cases which are of interest for classification theory, the total spaces of these bundles will clearly be projective. It is, however, generall…
Enumeration of Rational Cuspidal Curves via the WDVV equation
本文扩展 Kontsevich 递归公式思想，提出射影平面有理尖点曲线特征数的猜想公式，并基于此技术扩展计算具有 E6 奇点的有理四次曲线特征数。
Let N_d be the number of degree d, nodal, rational plane curves through 3d-1 points in the complex projective plane. The number of degree d>=3, nodal, elliptic plane curves with a fixed (general) j-invariant through 3d-1 points is found to be {d-1 \choose 2}*N_d.
On the topology of fiber-type curves: a Zariski pair of affine nodal curves
本文研究光滑射影曲面中曲线补空间基本群为循环群自由积的条件，构造无穷多纤维型曲线Zariski对，含复仿射平面内仅含结点的曲线对，证明结点位置影响射影曲线嵌入拓扑，并用有限SU(2)表示的扭曲Alexander多项式区分同阿贝尔不变量的基本群。
Let P^2_r be the projective plane blown up at r generic points. Denote by E_0,E_1,...,E_r the strict transform of a generic straight line on P^2 and the exceptional divisors of the blown-up points on P^2_r respectively. We consider the variety V of all irreducible curves C in |dE_0-d_1E_1-...-d_rE_r| with k nodes as the only singularities and give asymptotically nearly optimal sufficient conditions for its smoothness, irreducibility and non-emptyness. Moreover, we extend our conditions for the smoothness and the irreducibility on families of reducible curves. For r<10 we give the complete answ…
We prove a new patchworking theorem for singular algebraic curves, which states the following. Given a complex toric threefold $Y$ which fibers over ${\mathbb C}$ with a reduced reducible zero fiber $Y_0$ and other fibers $Y_t$ smooth, and given a reduced curve $C_0\subset Y_0$, the theorem provides a sufficient condition for the existence of a one-parametric family of curves $C_t\subset Y_t$, which induces an equisingular deformation for some singular points of $C_0$ and certain prescribed deformations for the other singularities. As application we give a comment on a recent theorem by G. Mik…
We study Pandharipande-Thomas's stable pair theory on $K3$ fibrations over curves with possibly nodal fibers. We describe stable pair invariants of the fiberwise irreducible curve classes in terms of Kawai-Yoshioka's formula for the Euler characteristics of moduli spaces of stable pairs on $K3$ surfaces and Noether-Lefschetz numbers of the fibration. Moreover, we investigate the relation of these invariants with the perverse (non-commutative) stable pair invariants of the $K3$ fibration. In the case that the $K3$ fibration is a projective Calabi-Yau threefold, by means of wall-crossing techniq…
Costantini and Kappes gave an algebraic equation of the universal family over the Kenyon-Smillie (2,3,4)-Teichmüller curve. This equation gives rise to a family of projective plane quartic curves with three singular members. These singular curves are: (1) an integral nodal quartic, (2) the union of a line and a nodal cubic, and (3) a quadruple line. We determine the limit Weierstrass points on these singular curves.
Newstead gave the generators of the cohomology ring of the moduli space of rank 2 semi-stable, torsion-free sheaves with fixed odd degree determinant over a smooth, projective curve. In this article, we generalize this result to the case when the underlying curve is irreducible, nodal. We show that these generators (of the cohomology ring in the nodal curve case) arise naturally as degeneration of Newstead's generators in the smooth curve case.
Fourier-Mukai transforms and normalisation of nodal curves
本文研究了阿林金的庞加莱sheaf在紧化雅可比簇奇异点上的性质，通过正则化得到庞加莱sheaf之间的关系，利用抛物模的模空间进行sheaf数据的交织，为后续研究奇点处的希奇金系统镜像对称提供基础。
Branched covers of the complex projective line ramified over $0,1$ and $\infty$ (Grothendieck's {\em dessins d'enfant}) of fixed genus and degree are effectively enumerated. More precisely, branched covers of a given ramification profile over $\infty$ and given numbers of preimages of $0$ and $1$ are considered. The generating function for the numbers of such covers is shown to satisfy a PDE that determines it uniquely modulo a simple initial condition. Moreover, this generating function satisfies an infinite system of PDE's called the KP (Kadomtsev-Petviashvili) hierarchy. A specification of …
Irreducibility of Some Quot Schemes on Nodal Curves
本文证明了对于算术亏格 $g \geqslant 2$ 的整射影节点曲线 $C$，当次数 $d$ 足够大时，向量丛 $E$ 的 Quot scheme $\textrm{Quot}_{C/\mathbb C}(E,k,d)$ 是不可约的。
Automorphic gluing functor in Betti Geometric Langlands
本文研究曲线结点退化中的幂零层自同构范畴，通过标记点鼓出射影直线解释仿射Hecke算子，并构造连接结点曲线与光滑化自同构范畴的粘合函子。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。