Parole chiave

• singular curve
• osculating space
• multifiltration

Abstract. How bad can singularities of a curve of degree $d$ in projective $n$-space be? The study of this question is very classical. Even for plane curves, all possible configurations of singularities are only known in low degrees. In higher dimensions, much less is known. In the eighties, Piene and Eisenbud-Harris studied flags of osculating spaces attached to linear series of curves. In this talk, we introduce a gadget (called multifiltration) obtained by combining those flags. We use it to give upper bounds on the arithmetic genus of projective curves in some ranges (reproving a result due to Castelnuovo). We classify all configurations of singularities that can arise when any smooth curve is projected from a linear space of dimension at most two. With these techniques, one can describe the Schubert cycles giving rise to those projections. This is joint work with J. Buczyński and N. Ilten.