International Journal of Mathematics

The "linear orbit" of a plane curve of degree d is its orbit in ℙ^d(d+3)/2 under the natural action of PGL(3). We classify curves with positive dimensional stabilizer, and we compute the degree of the closure of the linear orbits of curves supported on unions of lines. Together with the results of [3], this encompasses the enumerative geometry of all plane curves with small linear orbit. This information will serve elsewhere as an ingredient in the computation of the degree of the orbit closure of an arbitrary plane curve.

Let (X, 0) be the germ of a normal space of dimension n+1 with an isolated singularity at 0 and let f be a germ of holomorphic function with an isolated regularity at 0. We prove that the meromorphic extension of the current

This article treats various aspects of the geometry of the moduli of r-spin curves and its compactification . Generalized spin curves, or r-spin curves, are a natural generalization of 2-spin curves (algebraic curves with a theta-characteristic), and have been of interest lately because of the similarities between the intersection theory of these moduli spaces and that of the moduli of stable maps. In particular, these spaces are the subject of a remarkable conjecture of E. Witten relating their intersection theory to the Gelfand–Dikii (KdV_r) heirarchy. There is also a W-algebra conjecture for these spaces [16] generalizing the Virasoro conjecture of quantum cohomology.

For any line bundle on the universal curve over the stack of stable curves, there is a smooth stack of triples (X, ℒ, b) of a smooth curve X, a line bundle ℒ on X, and an isomorphism . In the special case that is the relative dualizing sheaf, then is the stack of r-spin curves.

We construct a smooth compactification of the stack , describe the geometric meaning of its points, and prove that its coarse moduli is projective.

We also prove that when r is odd and g>1, the compactified stack of spin curves and its coarse moduli space are irreducible, and when r is even and is the disjoint union of two irreducible components. We give similar results for n-pointed spin curves, as required for Witten's conjecture, and also generalize to the n-pointed case the classical fact that when is the disjoint union of d(r) components, where d(r) is the number of positive divisors of r. These irreducibility properties are important in the study of the Picard group of [15], and also in the study of the cohomological field theory related to Witten's conjecture [16, 34].

This paper studies the (small) quantum homology and cohomology of fibrations p:P→S² whose structural group is the group of Hamiltonian symplectomorphisms of the fiber (M, ω). It gives a proof that the rational cohomology splits additively as the vector space tensor product H*(M)⊗H*(S²), and investigates conditions under which the ring structure also splits, thus generalizing work of Lalonde–McDuff–Polterovich and Seidel. The main tool is a study of certain operations in the quantum homology of the total space P and of the fiber M, whose properties reflect the relations between the Gromov–Witten invariants of P and M. In order to establish these properties we further develop the language introduced in [22] to describe the virtual moduli cycle (defined by Liu–Tian, Fukaya–Ono, Li–Tian, Ruan and Siebert).

We give several Hartogs-type extension theorems for separately holomorphic mappings on compact sets into complex spaces which either have the Hartogs extension property or are weakly Brody hyperbolic. Moreover, a characterization for the Hartogs extension property of holomorphically convex Kähler complex spaces by separate analyticity is given.