We prove that some Gromov-Witten numbers associated to rational contact (Legendrian) curves in any contact complex projective space with arbitrary incidence conditions are enumerative. Also, we use Bott formula on the Kontsevich space to find the exact value of those numbers. As an example, the numbers of rational contact curves of low degree in $\mathbb{P}^{3}$ and $\mathbb{P}^{5}$ are computed. The results are consistent with existing results.
The invariance of the Welschinger numbers for real unnodal Del Pezzo surfaces, which we used for the enumeration of real rational curves on real toric Del Pezzo surfaces (see math.AG/0303378 and IMRN 49 (2003), 2639-2653), follows from J.-Y. Welschinger's theorem stated and proved in a symplectic setting (math.AG/0303145 and CRAS, Ser. I, 336 (2003), 341-344). Here, we give an algebraic geometry version of the proof of the above invariance.
We use twisted stable maps to answer the following question. Let E\subset P^2 be a smooth cubic. How many rational degree d curves pass through a general points of E, have b specified tangencies with E and c unspecified tangencies, and pass through 3d-1-a-2b-c general points of P^2? The answer is given as a generalization of Kontsevich's recursion. We also investigate more general enumerative problems of this sort, and prove an analogue of a formula of Caporaso and Harris.
In this paper we prove the algebraic-tropical correspondence for stable maps of rational curves with marked points to toric varieties such that the marked points are mapped to given orbits in the big torus and in the boundary divisor, the map has prescribed tangency to the boundary divisor, and certain quadruples of marked points have prescribed cross-ratios. In particular, our results generalize the results of Nishinou and Siebert. The proof is very short, involves only the standard theory of schemes, and works in arbitrary characteristic (including the mixed characteristic case).
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…
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.
We answer some enumerative questions about irreducible rational curves on Hirzebruch surfaces, by combining an idea of Kontsevich with the study of the geometry of certain natural parameter spaces. Our formulas generalize Kontsevich's formula for rational plane curves of given degree.
A smooth variety is called uniformly rational if every point admits a Zariski open neighborhood isomorphic to a Zariski open subset of the affine space. In this note we show that every smooth and rational affine variety endowed with an algebraic torus action such that the algebraic quotient has dimension 0 or 1 is uniformly rational.
Complex projective algebraic varieties with $\mathbb{C}^*$-actions can be thought of as geometric counterparts of birational transformations. In this paper we describe geometrically the birational transformations associated to rational homogeneous varieties endowed with a $\mathbb{C}^*$-action with no proper isotropy subgroups.
Enumerating ramified coverings of the sphere with fixed ramification types is a well-known problem first considered by A. Hurwitz. Up to now, explicit solutions have been obtained only for some families of ramified coverings, for instant, those realized by polynomials in one complex variable. In this paper we obtain an explicit answer for a large new family of coverings, namely, the coverings realized by simple almost polynomials, defined below. Unlike most other results in the field, our formula is obtained by elementary methods.
We study full rank homogeneous valuations on (multi)-graded domains and ask when they have finite Khovanskii bases. We show that there is a natural reduction from multigraded to simply graded domains. As special cases, we consider projective coordinate rings of rational curves, and almost toric varieties. Our results relate to several problems posed by Kaveh and Manon, and imply that the procedure of Bossinger-Lamboglia-Mincheva-Mohammadi for producing tropical prime cones will not terminate in general.
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…
One distinguishing feature of rational curves is that they have algebraic parameterizations. Arc spaces are a way of describing approximations to parameterizations of all curves in some fixed space. Playing on these descriptions, this paper introduces rationally generated arcs, and applies them towards the description and enumeration of rational curves in algebraic varieties.
A new proof of the mirror conjecture for Fano and Calabi-Yau complete intersections in P^n is given, using only the circle action on the graph space. The proof applies to projective bundles as well, with applications to "linear" relative Calabi-Yau's and to Schubert calculus.
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.
Comessatti proved that the set of real points of a rational real algebraic surface is either a nonorientable surface, or the two-sphere, or the torus. Conversely, it is easy to see that all of these surfaces admit a rational real algebraic model. We prove that they admit exactly one rational real algebraic model. This was known earlier only for the two-sphere, torus, projective plane and the Klein bottle.
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.
For any prime number $p$ and field $k$, we characterize the $p$-retract rationality of an algebraic $k$-torus in terms of its character lattice. We show that a $k$-torus is retract rational if and only if it is $p$-retract rational for every prime $p$, and that the Noether problem for retract rationality for a group of multiplicative type $G$ has an affirmative answer for $G$ if and only if the Noether problem for $p$-retract rationality for $G$ have a positive answer for all $p$. For every finite set of primes $S$ we give examples of tori that are $p$-retract rational if and only if $p\notin …
The logarithmic multiplicative group is a proper group object in logarithmic schemes, which morally compactifies the usual multiplicative group. We study the structure of the stacks of logarithmic maps from rational curves to this logarithmic torus, and show that in most cases, it is a product of the logarithmic torus with the space of rational curves. This gives a conceptual explanation for earlier results on the moduli spaces of logarithmic stable maps to toric varieties.
In this work we provide effective bounds and classification results for rational $\QQ$-factorial Fano varieties with a complexity-one torus action and Picard number one depending on the invariants dimension and Picard index. This complements earlier work by Hausen, Süß and the author, where the case of free divisor class group of rank one was treated.
Let X be a Mori dream space together with an effective torus action of complexity one. In this note, we construct a polyhedral divisor on a suitable covering of the projective line P^1 which corresponds to the affine spectrum of the Cox ring of X. This description allows for a detailed study of torus orbits and deformations of the latter. Moreover, we present coverings of P^1 together with an action of a finite abelian group A in terms of so-called A-divisors of degree zero on P^1.
In this paper I consider three families of orders on the set of r- multipartitions of n, depending on r rational parameters. The one we are interested in is a geometric order on the set of fixed points for a torus action on the Gieseker variety. The other two serve as bounds for the geometric one, providing an almost accurate description.
We study the stable pairs theory of local curves in 3-folds with descendent insertions. The rationality of the partition function of descendent invariants is established for the full local curve geometry (equivariant with respect to the scaling 2-torus) including relative conditions and odd degree insertions for higher genus curves. The capped 1-leg descendent vertex (equivariant with respect to the 3-torus) is also proven to be rational. The results are obtained by combining geometric constraints with a detailed analysis of the poles of the descendent vertex.
We study the equivariant cobordism rings for the action of a torus $T$ on smooth varieties over an algebraically closed field of characteristic zero. We prove a theorem describing the rational $T$-equivariant cobordism rings of smooth projective $G$-spherical varieties with the action of a maximal torus $T$ of $G$. As an application, we obtain explicit presentations for the rational equivariant cobordism rings of smooth projective horospherical varieties of Picard number one.
We address the problem of existence of refined (i.e., depending on a formal parameter) tropical enumerative invariants, and we present two new examples of a refined count of rational marked tropical curves. One of the new invariants counts plane rational tropical curves with an unmarked vertex of arbitrary valency. It was motivated by the tropical enumeration of plane cuspidal tropical curves given by Y. Ganor and the author, which naturally led to consideration of plane tropical curves with an unmarked four-valent vertex. Another refined invariant counts rational tropical curves of a given de…
We conjecturally extract the triply graded Khovanov-Rozansky homology of the (m, n) torus knot from the unique finite dimensional simple representation of the rational DAHA of type A, rank n - 1, and central character m/n. The conjectural differentials of Gukov, Dunfield and the third author receive an explicit algebraic expression in this picture, yielding a prescription for the doubly graded Khovanov-Rozansky homologies. We match our conjecture to previous conjectures of the first author relating knot homology to q, t-Catalan numbers, and of the last three authors relating knot homology to H…
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. …
We systematically produce algebraic varieties with torus action by constructing them as suitably embedded subvarieties of toric varieties. The resulting varieties admit an explicit treatment in terms of toric geometry and graded ring theory. Our approach extends existing constructions of rational varieties with torus action of complexity one and delivers all Mori dream spaces with torus action. We exhibit the example class of general arrangement varieties and obtain classification results in the case of complexity two and Picard number at most two, extending former work in complexity one.
Using Altmann-Hausen-Suss description of $\mathbb{T}$-varieties via divisorial fans and Kóvacs-Schwede-Smith characterization of Du Bois singularities, we study Cohen-Macaulay Du Bois $\mathbb{T}$-singularities of complexity one. We exhibit cohomological criteria for a $\mathbb{T}$-variety to be Cohen-Macaulay and Du Bois in terms of polyhedral divisors. We give an example of a Cohen-Macaulay Du Bois singularity of complexity one which does not have rational singularities.
We consider a normal complete rational variety with a torus action of complexity one. In the main results, we determine the roots of the automorphism group and give an explicit description of the root system of its semisimple part. The results are applied to the study of almost homogeneous varieties. For example, we describe all almost homogeneous (possibly singular) del Pezzo k*-surfaces of Picard number one and all almost homogeneous (possibly singular) Fano threefolds of Picard number one having a reductive automorphism group with two-dimensional maximal torus.
We compute rational equivariant Chow rings with respect to a torus of quiver moduli spaces. We derive a presentation in terms of generators and relations, use torus localization to identify it as a subring of the Chow ring of the fixed point locus, and we compare the two descriptions.
We provide a notion of algebraic rational cell with applications to intersection theory on singular varieties with torus action. Based on this notion, we study the algebraic analogue of $\mathbb{Q}$-filtrable varieties: algebraic varieties where a torus acts with isolated fixed points, such that the associated Bialynicki-Birula decomposition consists of algebraic rational cells. We show that the rational equivariant Chow group of any $\mathbb{Q}$-filtrable variety is freely generated by the cell closures. We apply this result to group embeddings, and more generally to spherical varieties. This…
Motivated by recent work in the mathematics and engineering literature, we study integrability and non-tangential regularity on the two-torus for rational functions that are holomorphic on the bidisk. One way to study such rational functions is to fix the denominator and look at the ideal of polynomials in the numerator such that the rational function is square integrable. A concrete list of generators is given for this ideal as well as a precise count of the dimension of the subspace of numerators with a specified bound on bidegree. The dimension count is accomplished by constructing a natura…
We construct a rational $T^2$-equivariant elliptic cohomology theory for the 2-torus $T^2$, starting from an elliptic curve C over the complex numbers and a coordinate data around the identity. The theory is defined by constructing an object $EC_{T^2}$ in the algebraic model category $dA(T^2)$, which by Greenlees and Shipley is Quillen-equivalent to rational $T^2$-spectra. This result is a generalization to the 2-torus of the construction [Gre05] for the circle. The object $EC_{T^2}$ is directly built using geometric inputs coming from the Cousin complex of the structure sheaf of the surface C…
We prove that if a Calabi--Yau manifold $M$ admits a holomorphic Cartan geometry, then $M$ is covered by a complex torus. This is done by establishing the Bogomolov inequality for semistable sheaves on compact Kähler manifolds. We also classify all holomorphic Cartan geometries on rationally connected complex projective manifolds.
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…
On intersection cohomology with torus action of complexity one, II
Basque Center of Applied Mathematics (BCAM)（巴斯克应用数学中心）；Ruhr-Universität Bochum（鲁尔大学波鸿分校）；Indian Institute of Technology Kanpur（坎普尔印度理工学院）；Universidade Federal do Ceará (UFC)（塞阿拉联邦大学）
