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

From optimal to practical safety standards for dike-ring areas

After the flood disaster in 1953 in the southwestern part of the Netherlands, Van Dantzig tried to solve the economic decision problem concerning the optimal height of dikes. His solution has a fixed probability of flooding after each investment (Econometrica, 1956). However, when there is economic growth, not the probability of flooding but the expected yearly loss by flooding is the key variable in the real optimal safety strategy. Under some conditions, it is optimal to keep this expected loss within a constant interval. Therefore, when the potential damage increases by economic growth, the flooding probability has to decline in the course of time in order to keep the expected loss between the fixed boundaries. The purpose of the paper is to show the implications of the optimal solution in case there are differences between costs and benefits among dike-ring areas. Further, the paper focuses on the translation of the theoretical results into new legal standards that can work well in practice. Cost benefit analysis, optimal height of dikes, optimal safety standards.

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.

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