Intersection-theoretical computations on \Mgbar

We determine necessary conditions for ample divisors in arbitrary genus as well as for very ample divisors in genus 2 and 3. We also compute the intersection numbers $\lambda^9$ and $\lambda_{g-1}^3$ in genus 4. The latter number is relevant for counting curves of higher genus on manifolds, cf. the recent work of Bershadsky et al.Comment: 13 pages, no figures. To appear in "Parameter Spaces", Banach Center Publications, volume in preparation. plain te

Limits of PGL(3)-translates of plane curves, II

Every complex plane curve C determines a subscheme S of the $P^8$ of 3x3 matrices, whose projective normal cone (PNC) captures subtle invariants of C. In "Limits of PGL(3)-translates of plane curves, I" we obtain a set-theoretic description of the PNC and thereby we determine all possible limits of families of plane curves whose general element is isomorphic to C. The main result of this article is the determination of the PNC as a cycle; this is an essential ingredient in our computation in "Linear orbits of arbitrary plane curves" of the degree of the PGL(3)-orbit closure of an arbitrary plane curve, an invariant of natural enumerative significance.Comment: 22 pages. Minor revision. Final versio

Linear orbits of arbitrary plane curves

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). In this paper we obtain an algorithm computing the degree of the closure of the linear orbit of an arbitrary plane curve, and give explicit formulas for plane curves with irreducible singularities. The main tool is an intersection@-theoretic study of the projective normal cone of a scheme determined by the curve in the projective space $\P^8$ of $3\times 3$ matrices; this expresses the degree of the orbit closure in terms of the degrees of suitable loci related to the limits of the curve. These limits, and the degrees of the corresponding loci, have been established in previous work.Comment: 33 pages, AmS-TeX 2.

The class of the bielliptic locus in genus 3

Let the bielliptic locus be the closure in the moduli space of stable curves of the locus of smooth curves that are double covers of genus 1 curves. In this paper we compute the class of the bielliptic locus in \bar{M}_3 in terms of a standard basis of the rational Chow group of codimension-2 classes in the moduli space. Our method is to test the class on the hyperelliptic locus: this gives the desired result up to two free parameters, which are then determined by intersecting the locus with two surfaces in \bar{M}_3.Comment: 14 pages, 2 figure