Given a smooth non-hyperelliptic curve C of genus 3 and a maximal isotropic subgroup (w.r.t. the Weil pairing) L in Jac(C)[2], there exists a smooth curve C' s.t. Jac(C')=Jac(C)/L. This construction is symmetric. i.e. if we start with C' and the dual flag on it, we get C. A previous less explicit approach was taken by Donagi and Livne. The advantage of our construction is that it is explicit enough to describe the isomorphism H^0(C,K_C)=H^0(C',K_C').
We discuss Weber's formula which gives the quotient of two Thetanullwerte for a plane smooth quartic in terms of the bitangents. In particular, we show how it can easily be derived from the Riemann-Jacobi formula.
The arithmetic geometric mean algorithm for calculation of elliptic integrals of the first type was introduced by Gauss. The analog algorithm for Abelian integrals of genus 2 was introduced by Richelot (1837) and Humbert (1901). We present the analogous algorithm for Abelian integrals of genus 3.
We compute the class of the closure of the locus of hyperelliptic curves in the moduli space of stable genus-3 curves in terms of the tautological class $λ$ and the boundary classes $δ_0$ and $δ_1$. The expression of this class is known, but here we compute it directly, by means of Porteous Formula, without resorting to blowups or test curves.
For local Calabi-Yau manifolds which are total spaces of vector bundle over balloon manifolds, we propose a formal definition of reduced Genus one Gromov-Witten invariants, by assigning contributions from the refined decorated rooted trees. We show that this definition satisfies a localized version of the standard versus reduced formula, whose global version in the compact cases is due to A. Zinger. As an application we prove the conjecture in a previous article on the genus one Gromov-Witten invariants of local Calabi-Yau manifolds which are total spaces of concave splitting vector bundles ov…
We obtain a formula for the degrees of the varieties parameterizing complex algebraic curves of any divisor class and genus on P^2_6, the projective plane blown-up at 6 generic points. Moreover, the formula computes the degrees of the varieties parameterizing curves on P^2_6 which additionally satisfy certain tangency conditions to a fixed exceptional divisor on P^2_6. Our formula contains as special cases the degrees of the analogous varieties parameterizing curves on P^2_q, for q=1,...,5, and on the quadric (P^1)^2. It is an extension of the Caporaso-Harris formula counting curves of any deg…
In this paper we construct infinitely many Shimura curves contained in the locus of Jacobians of genus four curves. All Jacobians in these families are ${\mathbb Z}/3$ covers of varying elliptic curves that appear in a geometric construction of Pirola, and include an example of a Shimura-Teichmuller curve that parameterizes Jacobians that are suitable ${\mathbb Z}/6$ covers of ${\mathbb P}^1$. We compute explicitly the period matrices of the Shimura curves we construct using the original construction of Shimura for moduli spaces of abelian varieties with automorphisms.
We give a formula for the number of genus-two fixed-complex-structure degree-d plane curves passing through 3d-2 points in general position. This is achieved by completing Katz-Qin-Ruan's approach. This paper's formula agrees with the one obtained by the author in a completely different way.
In this paper we extend the concept of Milnor fiber and Milnor number of a curve singularity allowing the ambient space to be a quotient surface singularity. A generalization of the local δ-invariant is defined and described in terms of a Q-resolution of the curve singularity. In particular, when applied to the classical case (the ambient space is a smooth surface) one obtains a formula for the classical δ-invariant in terms of a Q-resolution, which simplifies considerably effective computations. All these tools will finally allow for an explicit description of the genus formula of a curve def…
By using the infinitesimally marking point to break the loop in the localization calculation as Kim and Lho, and Zinger's explicit formulas for double $J$-functions, we obtain a formula for genus one stable quasimaps invariants when the target is a complete intersection Calabi-Yau in projective space, which gives a new proof of Kim and Lho's mirror theorem for elliptic quasimap invariants.
We provide an explicit formula for all primary genus-zero $r$-spin invariants. Our formula is piecewise polynomial in the monodromies at each marked point and in $r$. To deduce the structure of these invariants, we use a tropical realization of the corresponding cohomological field theories. We observe that the collection of all WDVV relations is equivalent to the relations deduced from the fact that genus-zero tropical CohFT cycles are balanced.
A relatively fast algorithm for evaluating Weil-Petersson volumes of moduli spaces of complex algebraic curves is proposed. On the basis of numerical data, a conjectural large genus asymptotics of the Weil-Petersson volumes is computed. Asymptotic formulas for the intersection numbers involving $ψ$-classes are conjectured as well. The accuracy of the formulas is high enough to believe that they are exact.
We prove a genus formula for modular curves of $D$-elliptic sheaves. We use this formula to show that the reductions of modular curves of $D$-elliptic sheaves attain the Drinfeld-Vladut bound as the degree of the discriminant of $D$ tends to infinity.
Effective reconstruction formulas of a curve from its theta hyperplanes are known classically in genus 2 (where the theta hyperplanes are Weierstrass points), and 3 (where, for a generic curve, the theta hyperplanes are bitangents to a plane quartic). However, for higher genera, no formula or algorithm are known. In this paper we give an explicit (and simple) algorithm for computing a generic genus 4 curve from it's theta hyperplanes.
I prove a formula expressing the descendent genus g Gromov-Witten invariants of a projective variety X in terms of genus 0 invariants of its symmetric product stack S^{g+1}(X). When X is a point, the latter are structure constants of the symmetric group, and we obtain a new way of calculating the Gromov-Witten invariants of a point.
Using the degeneration formula and absolute/relative correspondence, one studied the change of Gromov-Witten invariants under blow-up for six dimensional symplectic manifolds and obtained closed blow-up formulae for high genus Gromov-Witten invariants. Our formulae also imply some relations among generalized BPS numbers introduced by Pandharipande.
We conjecture a formula for the generating function of genus one Gromov-Witten invariants of the local Calabi-Yau manifolds which are the total spaces of splitting bundles over projective spaces. We prove this conjecture in several special cases, and assuming the validity of our conjecture we check the integrality of genus one BPS numbers of local Calabi-Yau 5-folds defined by A. Klemm and R. Pandharipande.
We discuss a family of multi-term addition formulae for Weierstrass functions on specialized curves of genus one and two with many automorphisms. In the genus one case we find new addition formulae for the equianharmonic and lemniscate cases, and in genus two we find some new addition formulae for a number of curves, including the Burnside curve.
We collect various known results (about plane curves and the moduli space of stable maps) to derive new recursive formulas enumerating low genus plane curves of any degree with various behaviors. Recursive formulas are given for the characteristic numbers of rational plane curves, elliptic plane curves, and elliptic plane curves with fixed complex structure. Recursions are also given for the number of elliptic (and rational) plane curves with various "codimension 1" behavior (cuspidal, tacnodal, triple pointed, etc., as well as the geometric and arithmetic sectional genus of the Severi variety…
For a Fermat quasi-homogeneous polynomial, we study the associated weighted Fan-Jarvis-Ruan-Witten theory with narrow insertions. We prove a wall-crossing formula in all genera via localization on a master space, which is constructed by introducing an additional tangent vector to the moduli problem. This is a Landau-Ginzburg theory analogue of the higher-genus quasi-map wall-crossing formula proved by Ciocan-Fontanine and Kim. It generalizes the genus-$0$ result by Ross-Ruan and the genus-$1$ result by Guo-Ross.
In this paper, we study the all genus Gromov-Witten theory for any GKM orbifold $X$. We generalize the Givental formula which is studied in the smooth case in \cite{Giv2} \cite{Giv3} \cite{Giv4} to the orbifold case. Specifically, we recover the higher genus Gromov-Witten invariants of a GKM orbifold $X$ by its genus zero data. When $X$ is toric, the genus zero Gromov-Witten invariants of $X$ can be explicitly computed by the mirror theorem studied in \cite{CCIT} and our main theorem gives a closed formula for the all genus Gromov-Witten invariants of $X$. When $X$ is a toric Calabi-Yau 3-orbi…
In previous work (arXiv:1304.7056) we have conjectured wall-crossing formulas for genus zero quasimap invariants of GIT quotients and proved them via localization in many cases. We extend these formulas to higher genus when the target is semi-positive, and prove them for semi-positive toric varieties, in particular for toric local Calabi-Yau targets. The proof also applies to local Calabi-Yau's associated to some non-abelian quotients.
We revisit the work of Chang and Ran on bounding the slopes of $\overline{\mathscr{M}_{15}}$ and $\overline{\mathscr{M}_{16}}$, correct one of the formulas used at the conclusion of the argument, and recompute the lower bounds on the slopes, yielding $s(\overline{\mathscr{M}_{15}})>6.5$ but not for $\overline{\mathscr{M}_{16}}$. Our contribution only involves plugging in formulas.
This is the continuation of our paper [10]. In this paper which is self contained, we would like to give a different obstruction formula to the FIRST order deformation of the pair of a smooth curve and a smooth hypersurface. This obstruction formula leads to a genus formula for a smooth curve in a smooth hypersurface. As an application we show that smooth elliptic curves in a smooth hypersurface of degree $$h\geq 2n-1$$ in the projective space $\mathbf P^n, n\geq 3,$ can't deform in the first order to all hypersurfaces of the same degree. In particular, there are no smooth elliptic curves in g…
We compute Gromov-Witten invariants of any genus for del Pezzo surfaces of degree $\ge2$. The genus zero invariants have been computed a long ago, Gromov-Witten invariants of any genus for del Pezzo surfaces of degree $\ge3$ have been found by Vakil. We solve the problem in two steps: (1) we consider the plane blown up at 6 points on a conic and one more point outside this conic (weak Fano surface), and, using techniques of tropical geometry, obtain a Caporaso-Harris type formula counting curves of any divisor class and genus subject to arbitrary tangency conditions with respect to the blown u…
A formula for the generating function of the Weil-Petersson volumes of moduli spaces of pointed curves that is identical to the genus expansion of the free energy in two dimensional gravity is obtained. The contribution of arbitrary genus is expressed in terms of the Bessel function $J_0$.
We define a family of arithmetic zero cycles in the arithmetic Chow group of a modular curve X_0(N), for N>3 odd and squarefree, and identify the arithmetic degrees of these cycles as q-coefficients of the central derivative of a Siegel Eisenstein series of genus two. This parallels work of Kudla-Rapoport-Yang for Shimura curves.
We derive a closed-form expression for all genus 1 Hurwitz numbers, and give a simple new graph-theoretic interpretation of Hurwitz numbers in genus 0 and 1. (Hurwitz numbers essentially count irreducible genus g covers of the sphere, with arbitrary specified branching over one point, simple branching over other specified points, and no other branching. The problem is equivalent to counting transitive factorisations of permutations into transpositions.) These results prove a conjecture of Goulden and Jackson, and extend results of Hurwitz and many others.
In this paper we generalize the formula of Frobenius-Stickelberger and the formula of Kiepert to genus-three case. The latter is well-known determinant expression for any division polynomial of any elliptic curve.
We classify global surfaces of section for the Reeb flow of the standard contact form on the 3-sphere, defining the Hopf fibration. As an application, we prove the degree-genus formula for complex projective curves, using an elementary degeneration process inspired by the language of holomorphic buildings in symplectic field theory.
We present a quasi-linear algorithm to compute isogenies between Jacobians of curves of genus 2 and 3 starting from the equation of the curve and a maximal isotropic subgroup of the l-torsion, for l an odd prime number, generalizing the Vélu's formula of genus 1. This work is based from the paper "Computing functions on Jacobians and their quotients" of Jean-Marc Couveignes and Tony Ezome. We improve their genus 2 case algorithm, generalize it for genus 3 hyperelliptic curves and introduce a way to deal with the genus 3 non-hyperelliptic case, using algebraic theta functions.
We obtain mirror formulas for the genus 1 Gromov-Witten invariants of projective Calabi-Yau complete intersections. We follow the approach previously used for projective hypersurfaces by extending the scope of its algebraic results; there is little change in the geometric aspects. As an application, we check the genus 1 BPS integrality predictions in low degrees for all projective complete intersections of dimensions 3, 4, and 5.
We give explicit formulas for the Kawazumi-Zhang invariant and Faltings delta-invariant of a compact and connected Riemann surface of genus three. The formulas are in terms of two integrals over the associated jacobian, one integral involving the standard Riemann theta function, and another involving a theta function particular to genus three that was discovered by Frobenius. We review part of Frobenius' work on his theta function and connect our results with a formula due to Bloch, Hain and Bost describing the archimedean height pairing of Ceresa cycles in genus three.
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…
We prove a general formula for the intersection form of two arbitrary monomials in boundary divisors. Furthermore we present a tree basis of the cohomology of $\overline {M}_{0,n}$. With the help of the intersection form we determine the Gram matrix for this basis and give a formula for its inverse. This enables us to calculate the tensor product of the higher order multiplications arising in quantum cohomology and formal Frobenius manifolds. In the context of quantum cohomology this establishes the explicit Künneth formula.
We obtain a recursive formula answering the following question: How many irreducible, plane curves of degree d and (geometric) genus g pass through 3d-1+g general points in the plane? The formula is proved by studying suitable degenerations of plane curves.
We give an explicit formula for the difference between the standard and reduced genus-one Gromov-Witten invariants. Combined with previous work on geometric properties of the latter, this paper makes it possible to compute the standard genus-one GW-invariants of complete intersections. In particular, we obtain a closed formula for the genus-one GW-invariants of a Calabi-Yau projective hypersurface and verify a recent mirror symmetry prediction for a sextic fourfold as a special case.
In this paper, we give a new genus-3 topological recursion relation for Gromov-Witten invariants of compact symplectic manifolds. This formula also applies to intersection numbers on moduli spaces of spin curves. A by-product of the proof of this formula is a new relation in the tautological ring of the moduli space of 1-pointed genus-3 stable curves.
We give a closed formula for recovering a non-hyperelliptic genus three curve from its period matrix, and derive some identities between Jacobian Nullwerte in dimension three.
Following Brooks's calculation of the $\hat{A}$-genus of complete intersections, a new and more computable formula about the $\hat{A}$-genus and $α$-invariant will be described as polynomials of multi-degree and dimension. We also give an iterated formula of $\hat{A}$-genus and the necessary and sufficient conditions for the vanishing of $\hat{A}$-genus of complex even dimensional spin complete intersections. Finally, we obtain a general formula about the Hirzebruch genus of complete intersections, and calculate some classical Hirzebruch genera as examples.
In this paper, we give a simple formula for the generating function of genus-2 Gromov-Witten invariants for manifolds with semisimple quantum cohomology, and use this formula to prove the genus-2 Virasoro conjecture for such manifolds.
We prove an explicit formula for the genus-one Fan-Jarvis-Ruan-Witten invariants associated to the quintic threefold, verifying the genus-one mirror conjecture of Huang, Klemm, and Quackenbush. The proof involves two steps. The first step uses localization on auxiliary moduli spaces to compare the usual Fan-Jarvis-Ruan-Witten invariants with a semisimple theory of twisted invariants. The second step uses the genus-one formula for semisimple cohomological field theories to compute the twisted invariants explicitly.
We present an efficient endomorphism for the Jacobian of a curve $C$ of genus 2 (hyperelliptic) for divisors having a Non disjoint support. This extends the work of Costello and Lauter in [12] who calculated explicit formulae for divisor doubling and addition of divisors with disjoint support in $\mathbb{J}(C)$ using only base field operations. Explicit formulae is presented for this third case and a slightly different approach for divisor doubling.
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.
Hassett spaces are moduli spaces of weighted stable pointed curves. In this work, we consider such spaces of curves of genus $0$ with weights all $\frac{1}{2}$. These spaces are interesting as they are isomorphic to $\overline{M}_{0,n}$ but have different universal families and different intersection theory. We develop a closed formula for intersections of $ψ$-classes on such spaces. In our main result, we encode the formula for top intersections in a generating function obtained by applying a differential operator to the Witten-potential.
Let $K$ be a function field over a finite field $k$ of characteristic $p$ and let $K_{\infty}/K$ be a geometric extension with Galois group $\mathbb{Z}_p$. Let $K_n$ be the corresponding subextension with Galois group $\mathbb{Z}/p^n\mathbb{Z}$ and genus $g_n$. In this paper, we give a simple explicit formula $g_n$ in terms of an explicit Witt vector construction of the $\mathbb{Z}_p$-tower. This formula leads to a tight lower bound on $g_n$ which is quadratic in $p^n$. Furthermore, we determine all $\mathbb{Z}_p$-towers for which the genus sequence is stable, in the sense that there are $a,b,…
The paper is devoted to the open topological recursion relations in genus 1, which are partial differential equations that conjecturally control open Gromov-Witten invariants in genus 1. We find an explicit formula for any solution analogous to the Dijkgraaf-Witten formula for a descendent Gromov-Witten potential in genus 1. We then prove that at the approximation up to genus 1 the exponent of an open descendent potential satisfies a system of explicitly constructed linear evolutionary PDEs with one spatial variable.
We completely characterize genus-0 K-theoretic Gromov-Witten invariants of a compact complex algebraic manifold in terms of cohomological Gromov-Witten invariants of this manifold. This is done by applying (a virtual version of) the Kawasaki-Hirzebruch-Riemann-Roch formula for expressing holomorphic Euler characteristics of orbibundles on moduli spaces of genus-0 stable maps, analyzing the sophisticated combinatorial structure of inertia stacks of such moduli spaces, and employing various quantum Riemann--Roch formulas from "fake" (i.e. orbifold-ignorant) quantum K-theory of manifold and orbif…
We derive a closed formula for the generating function of genus two Gromov-Witten invariants of quintic 3-folds and verify the corresponding mirror symmetry conjecture of Bershadsky, Cecotti, Ooguri and Vafa.
We prove a localization formula for virtual fundamental classes in the context of torus equivariant perfect obstruction theories. As an application, the higher genus Gromov-Witten invariants of projective space are expressed as graph sums of tautological integrals over moduli spaces of stable pointed curves (generalizing Kontsevich's genus 0 formulas). Also, excess integrals over spaces of higher genus multiple covers are computed.
Let X be a proper smooth and connected surface over a finite field. We proved in [LLR2] that the order of the Brauer group Br(X) of X is a perfect square if it is finite. Our proof is based in part on a result of Gordon [Gor], which we used in [LLR1] to establish a key formula. Thomas Geisser noted that the formula in [LLR1] is incorrect, due to an omission in [Gor], and provides a corrected formula. We explain in this corrigendum how to modify the work of Gordon to establish a correct formula. The corrected formula can be used to prove the result in [LLR2] without further modifications.
D. Abramovich found an error in the singularity analysis in the first posting of this paper affecting one formula (Thanks Dan).The error has been corrected. Only the arithmetic genus formula has changed. The geometric genus and all the canonical class formulas are the same.
