A fundamental problem in algebraic geometry is the study of the nature of the moduli space M(g) of curves of genus g as an algebraic variety. It is known that for low g this space behaves very much like a projective space which means that one can describe explicitly most curves of given genus. However for g large the nature of M(g) changes completely and the investigator studies this transition by constructing new loci in the moduli space, which are of a completely different nature than those studied by previous mathematicians. The geometry of these new loci strongly suggests that the transition in the nature of M(g) appears much earlier than the leading conjecture in the field has predicted. The technical machinery developed by the investigator to solve this problem is also employed to study some open problems about vector bundles on general curves of genus g. A different project concerns the moduli space of pointed curves of genus 0. Obects of striking beauty, these spaces are important because of their pivotal role in new developments in enumerative geometry and because often statements about curves of arbitrary genus can be reduced to questions about pointed curves of genus 0. The investigator studies several invariants of these spaces and the proposed approach has been succesful in low dimensional cases.
Algebraic curves are ubiquitous objects in mathematics. They appear in complex analysis (as Riemann surfaces), in algebra (as field extensions) and in algebraic geometry (as curves in projective spaces). The most important problem in algebraic geometry is to classify algebraic varieties. For one-dimensional varieties (that is, for algebraic curves) this problem is approached by considering the moduli space M(g) of all curves of given genus. This is the universal parameter space for curves of fixed genus and interest in its geometry comes from fields as diverse as algebraic geometry, theoretical physics, number theory or coding theory. By understanding this space not only can we learn about families of curves but quite often the moduli space itself has beautiful geometric and arithmetic properties which one cannot expect to find on an arbitrary space.
