The delta invariant of curves on rational surfaces II: Poincar\'e series and topological aspects

This is a survey of the rationality problem in invariant theory. It also contains some new results, in particular in Chapter 2 on moduli spaces of plane curves with a theta-characteristic, and a detailed account of the relation of the Hesselink stratification of the Hilbert nullcone to the rationality problem, with an application to the rationality of the moduli space of plane curves of degree 34.

I give a conjectural generating function for the numbers of $\delta$-nodal curves in a linear system of dimension $\delta$ on an algebraic surface. It reproduces the results of Vainsencher for the case $\delta\le 6$ and Kleiman-Piene for the case $\delta\le 8$. The numbers of curves are expressed in terms of five universal power series, three of which I give explicitly as quasimodular forms. This gives in particular the numbers of curves of arbitrary genus on a K3 surface and...

We consider families of schemes over arbitrary fields resp. analytic varieties with finitely many (not necessarily reduced) isolated non-normal singularities, in particular families of generically reduced curves. We define a modified delta invariant for isolated non-normal singularities of any dimension that takes care of embedded points and prove that it behaves upper semicontinuous in flat families parametrized by an arbitrary principal ideal domain. Moreover, if the fibers...

In this paper, we will give a precise formula to compute delta invariants of projective bundles and projective cones of Fano type.

In a previous paper, the authors introduced a filtration on the ring ${\cal O}_{V,0}$ of germs of functions on a germ $(V,0)$ of a complex analytic variety defined by arcs on the singularity and called the arc filtration. The Poincar\'e series of this filtration were computed for simple surface singularities in the 3-space. Here they are computed for surface singularities from Arnold's lists including uni- and bimodular ones. The classification of the unimodular singularities...

The Euler characteristic of the link of a real algebraic variety is an interesting topological invariant in order to discuss local topological properties. We prove in the paper that an invariant stronger than the Euler Characteristic is well defined for the link of an algebraic variety: its virtual Poincar\'e polynomial.

The Hirzebruch $\chi_y$-genus and Poincare polynomial share some similar features. In this article we investigate two of their similar features simultaneously. Through this process we shall derive several new results as well as reprove and improve some known results.

This paper deals with a complete invariant $R_X$ for cyclic quotient surface singularities. This invariant appears in the Riemann Roch and Numerical Adjunction Formulas for normal surface singularities. Our goal is to give an explicit formula for $R_X$ based on the numerical information of $X$, that is, $d$ and $q$ as in $X=X(d;1,q)$. In the process, the space of curvettes and generic curves is explicitly described. We also define and describe other invariants of curves in $X...

The development of computational techniques in the last decade has made possible to attack some classical problems of algebraic geometry. In this survey, we briefly describe some open problems related to algebraic curves which can be approached from a computational viewpoint.

This paper surveys recent progress towards the Manin conjecture for (singular and non-singular) del Pezzo surfaces. To illustrate some of the techniques available, an upper bound of the expected order of magnitude is established for a singular del Pezzo surface of degree four.