Linear orbits of plane curves

Paolo Aluffi (Florida State University)

In how many ways can one realize a general genus-3 smooth curve as a plane curve, subject to the condition of containing 8 general points? One may view this problem as an enumerative question concerning smooth plane quartic curves; the same question may be asked for plane curves of arbitrary degree and with arbitrary singularities. It may be addressed by analysing the orbit of a given plane curve under the natural PGL(3) action. In joint work with Carel Faber, we study the enumerative geometry of these ‘linear orbits’, and obtain a complete answer to the original question for arbitrary curves, in terms of both global and local invariants of the curve. In the lectures we will summarize the necessary intersection theoretic background, and describe in some detail the constructions leading to a solution of this enumerative problem.