HITCHIN SYSTEMS ON SINGULAR CURVES II: GLUING SUBSCHEMES

Hitchin systems on ll-curves

Gluing pseudoholomorphic curves along branched covered cylinders II

This paper and its prequel (“Part I”) prove a generalization of the usual gluing theorem for two index 1 pseudoholomorphic curves U+ and U− in the symplectization of a contact 3-manifold. We assume that for each embedded Reeb orbit γ, the total multiplicity of the negative ends of U+ at covers of γ agrees with the total multiplicity of the positive ends of U− at covers of γ. However, unlike in ...

Enumerating singular curves on surfaces

We enumerate the singular algebraic curves in a complete linear system on a smooth projective surface. The system must be suitably ample in a rather precise sense. The curves may have up to eight nodes, or a triple point of a given type and up to three nodes. The curves must also pass through appropriately many general points. The number of curves is given by a universal polynomial in four basi...

Singular constrained linear systems

In the linear system Ax = b the points x are sometimes constrained to lie in a given subspace S of column space of A. Drazin inverse for any singular or nonsingular matrix, exist and is unique. In this paper, the singular consistent or inconsistent constrained linear systems are introduced and the effect of Drazin inverse in solving such systems is investigated. Constrained linear system arise ...

Gluing Pseudoholomorphic Curves along Branched Covered Cylinders I 1 Statement of the Gluing Theorem 1.1 Pseudoholomorphic Curves in Symplectizations

This paper and its sequel prove a generalization of the usual gluing theorem for two index 1 pseudoholomorphic curves u+ and u− in the symplectization of a contact 3-manifold. We assume that for each embedded Reeb orbit γ, the total multiplicity of the negative ends of u+ at covers of γ agrees with the total multiplicity of the positive ends of u− at covers of γ. However, unlike in the usual gl...