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Register a new userEnumeration of One-Nodal Rational Curves in Projective Spaces
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Authors
A. Zinger
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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.
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In our previous work, we provided an algebraic proof of the Zingers comparison formula between genus one Gromov-Witten invariants and reduced invariants when the target space is a complete intersection of dimension two or three in a projective space. In this paper, we extend the result in any dimensions and for descendant invariants.
Let $M$ be a complex manifold. We prove that a compact submanifold $Ssubset M$ with splitting tangent sequence (called a splitting submanifold) is rational homogeneous when $M$ is in a large class of rational homogeneous spaces of Picard number one. Moreover, when $M$ is irreducible Hermitian symmetric, we prove that $S$ must be also Hermitian symmetric. The basic tool we use is the restriction and projection map $pi$ of the global holomorphic vector fields on the ambient space which is induced from the splitting condition. The usage of global holomorphic vector fields may help us set up a new scheme to classify the splitting submanifolds in explicit examples, as an example we give a differential geometric proof for the classification of compact splitting submanifolds with $dimgeq 2$ in a hyperquadric, which has been previously proven using algebraic geometry.
For a Lagrangian torus A in a simply-connected projective symplectic manifold M, we prove that M has a hypersurface disjoint from a deformation of A. This implies that a Lagrangian torus in a compact hyperkahler manifold is a fiber of an almost holomorphic Lagrangian fibration, giving an affirmative answer to a question of Beauvilles. Our proof employs two different tools: the theory of action-angle variables for algebraically completely integrable Hamiltonian systems and Wielandts theory of subnormal subgroups.
Let $C$ be an algebraic curve of genus $g$ and $L$ a line bundle over $C$. Let $mathcal{MS}_C(n,L)$ and $mathcal{MO}_C(n,L)$ be the moduli spaces of $L$-valued symplectic and orthogonal bundles respectively, over $C$ of rank $n$. We construct rational curves on these moduli spaces which generalize Hecke curves on the moduli space of vector bundles. As a main result, we show that these Hecke type curves have the minimal degree among the rational curves passing through a general point of the moduli spaces. As its byproducts, we show the non-abelian Torelli theorem and compute the automorphism group of moduli spaces.
We provide an algorithm to check whether two rational space curves are related by a similarity. The algorithm exploits the relationship between the curvatures and torsions of two similar curves, which is formulated in a computer algebra setting. Helical curves, where curvature and torsion are proportional, need to be distinguished as a special case. The algorithm is easy to implement, as it involves only standard computer algebra techniques, such as greatest common divisors and resultants, and Grobner basis for the special case of helical curves. Details on the implementation and experimentation carried out using the computer algebra system Maple 18 are provided.
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