Anita Buckley

Ice cream and orbifold Riemann-Roch

We give an orbifold Riemann-Roch formula in closed form for the Hilbert series of a quasismooth polarized n-fold (X,D), under the assumption that X is projectively Gorenstein with only isolated orbifold points. Our formula is a sum of parts each of which is integral and Gorenstein symmetric of the same canonical weight; the orbifold parts are called ice cream functions. This form of the Hilbert series is particularly useful for computer algebra, and we illustrate it on examples of K3 surfaces and Calabi-Yau 3-folds. These results apply also with higher dimensional orbifold strata (see [1] and [2]), although the precise statements are considerably trickier. We expect to return to this in future publications.

Linear systems of plane curves with a composite number of base points of equal multiplicity

In this article we study linear systems of plane curves of degree d passing through general base points with the same multiplicity at each of them. These systems are known as homogeneous linear systems. We especially investigate for which of these systems, the base points, with their multiplicities, impose independent conditions and which homogeneous systems are empty. Such systems are called non-special. We extend the range of homogeneous linear systems that are known to be non-special. A theorem of Evain states that the systems of curves of degree d with 4 h base points with equal multiplicity are non-special. The analogous result for 9 h points was conjectured. Both of these will follow, as corollaries, from the main theorem proved in this paper. Also, the case of 4 h 9 k points will follow from our result. The proof uses a degeneration technique developed by C. Ciliberto and R. Miranda.

Orbifold Riemann–Roch for threefolds with an application to Calabi–Yau geometry

We prove an orbifold Riemann–Roch formula for a polarized complex 3– fold (X, D). As an application, we construct new families of projective Calabi–Yau threefolds.

Explicit determinantal representations of up to quintic bivariate polynomials

Abstract For bivariate polynomials of degree we give fast numerical constructions of determinantal representations with matrices. Unlike some other existing constructions, our approach returns matrices of the smallest possible size for all (not just generic) polynomials of degree n and does not require any symbolic computation. We can apply these linearizations to numerically compute the roots of a system of two bivariate polynomials by using numerical methods for the two-parameter eigenvalue problems.