Catriona Maclean

Vanishing sequences and Okounkov bodies

We define and study the vanishing sequence along a real valuation of sections of a line bundle on a normal projective variety. Building on previous work of the first author with Huayi Chen, we prove an equidistribution result for vanishing sequences of large powers of a big line bundle, and study the limit measure; in particular, the latter is described in terms of restricted volumes for divisorial valuations. We also show on an example that the associated concave function on the Okounkov body can be discontinuous at boundary points.

Explicit Noether-Lefschetz for arbitrary threefolds

We study the Noether-Lefschetz locus of a very ample line bundle L on an arbitrary smooth threefold Y . Building on results of Green, Voisin and Otwinowska, we give explicit bounds, depending only on the CastelnuovoMumford regularity properties of L, on the codimension of the components of the Noether-Lefschetz locus of jLj .

Deformations of locally complete intersections

Given a projective l.c.i. scheme, X⊂PN, we show that X has a smooth formal neighbourhood in which X is globally a complete intersection; that is, X is the intersection of codim(X) hypersurfaces. To cite this article: C. Maclean, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 355–358.

Random Finite Topologies and their Thresholds

For each integer n, there is a natural family of probability distributions on the set of topologies on a set of n elements, parametrized by an integer variable, m. We will describe how these are constructed and analysed, and find threshold functions (for m in terms of n) for various topological properties; we focus attention on connectivity and the size of the largest component.