Number Of Rational Curves Research Articles

Enumerative geometry of elliptic curves on toric surfaces

We establish the equality of classical and tropical curve counts for elliptic curves on toric surfaces with fixed $j$-invariant, refining results of Mikhalkin and Nishinou--Siebert. As an application, we determine a formula for such counts on $\mathbb P^2$ and all Hirzebruch surfaces. This formula relates the count of elliptic curves with the number of rational curves on the surface satisfying a small number of tangency conditions with the toric boundary. Furthermore, the combinatorial tropical multiplicities of Kerber and Markwig for counts in $\mathbb P^2$ are derived and explained algebro-geometrically, using Berkovich geometry and logarithmic Gromov--Witten theory. As a consequence, a new proof of Pandharipande's formula for counts of elliptic curves in $\mathbb P^2$ with fixed $j$-invariant is obtained.

Rational curves on Calabi–Yau threefolds: Verifying mirror symmetry predictions

In this paper, the numbers of rational curves on general complete intersection Calabi–Yau threefolds in complex projective spaces are computed up to degree six. The results are all in agreement with the predictions made from mirror symmetry.

The Number of Rational Curves on K3 Surfaces

Let X be a K3 surface with a primitive ample divisor H, and let $\beta=2[H]\in H_2(X, \mathbf Z)$. We calculate the Gromov-Witten type invariants $n_{\beta}$ by virtue of Euler numbers of some moduli spaces of stable sheaves. Eventually, it verifies Yau-Zaslow formula in the non primitive class $\beta$.

On Lang's Conjecture for Surfaces of General Type

Let m be a nonnegative integer. We explicitly bound the number of rational curves with arithmetic genus no greater than m on a surface of general type by geometric invariants of the surface. We also briefly discuss the possibility of bounding all rational curves lying on a surface of general type in ℙ3.

CALCULATION OF GROMOV-WITTEN INVARIANTS FOR CP3, CP4AND Gr(2, 4)

Using the associativity relations of the topological sigma models with target spaces, CP3, CP4 and Gr (2, 4), we derive recursion relations of their correlation and evaluate them up to a certain order in the expansion over the instantons. The expansion coefficients are regarded as the number of rational curves in CP3, CP4 and Gr (2, 4) which intersect various types of submanifolds corresponding to the choice of BRST-invariant operators in the correlation functions.

The Calabi-Yau moduli and the Toda equations

For a one-parameter family of Calabi-Yau d-fold embedded in C P d+1 we analyze the hermitian metrics on the moduli space of the non-linear sigma model by topological field techniques. The set of equations turns out to be the non-affine A-type Toda equation. It is contrasted with the C P d+1 case where the affine Toda equation appears. We obtain exact solutions of the hermitian metrics and associate their non-perturbative corrections to the holomorphic maps. In particular for the quintic case, the coefficients in the q-expansion are given by the number of rational curves. From these data, we construct the genus one partition function F 1 of the topological sigma model by the method of the holomorphic anomaly. In the asymmetrical limit of the Kähler parameter t → ∞ , the contribution from the A ∗- part is decoupled and we obtain the partition function of A-matter coupled with topological gravity at the stringy one-loop level. The coefficients of the series expansion are related with the integrals of the top Chem class of some vector bundle over the stable maps with definite degrees. We also comment on the behavior of the F 1 near the conifold locus.

Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties

We formulate general conjectures about the relationship between the A-model connection on the cohomology of ad-dimensional Calabi-Yau complete intersectionV ofr hypersurfacesV 1 ,...,V r in a toric varietyP Σ and the system of differential operators annihilating the special generalized hypergeometric series Φ0 constructed from the fan Σ. Using this generalized hypergeometric series, we propose conjectural mirrorsV′ ofV and the canonicalq-coordinates on the moduli spaces of Calabi-Yau manifolds.In the second part of the paper we consider some examples of Calabi-Yau 3-folds having Picard number >1 in products of projective spaces. For conjectural mirrors, using the recurrent relation among coefficients of the restriction of the hypergeometric function Φ0 on a special line in the moduli space, we determine the Picard-Fuchs equation satisfied by periods of this special one-parameter subfamily. This allows to obtain some sequences of integers which can be conjecturally interpreted in terms of Gromov-Witten invariants. Using standard techniques from enumerative geometry, first terms of these sequence of integers are checked to coincide with numbers of rational curves on Calabi-Yau 3-folds.