Caporaso 9801465 The goal of this project is to solve the following classical conjecture in algebraic geometry: the linear system of plane curves of fixed degree having a given number of singularities at assigned general points in the plane has dimension equal to the expected value. Such an expected value can be computed naively, and the above problem has been an object of study since the beginning of this century. Work that addresses this question would lead to a better understanding of the geometry of moduli spaces parameterizing plane curves, which are basic objects of study both in classical and modern algebraic geometry. The PI expects that degeneration techniques can be successfully used to approach these issues; in fact modern tools (such as semistable reduction and deformation theory) make degeneration methods much more powerful today than they were ever in the past. There are many open problems regarding the geometry of curves on algebraic varieties, of which the one described here is an example. It often happens that breakthroughs on one of them shed light on others. This project is an example of this fact, as the approach to be used here contains ideas that were used by the PI to prove results in the context of enumerative algebraic geometry. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.
