Variety of singular quadrics containing a projective curve (Q2326262)

Fix integers \(r\ge 3\), \(g\ge 0\) and \(d\) such that \(\rho (g,r,d):= (r+1)d- rg -r(r+1)\ge 0\). It is well-know that there is a unique irreducible component \(\Gamma\) of the Hilbert scheme of smooth curves \(C\subset \mathbb {P}^r\) parametrizing curves with general moduli and that a general \(C\in \Gamma\) has maximal rank in the range of quadrics, i.e. \(h^0(\mathcal{I} _C(2)) =\max \{0,\binom{r+2}{2} -2d-1+g\}\). In this paper the author study the stratification by ranks of the set of quadric hypersurfaces containing \(C\), giving a conjectural number of the dimension with fixed rank and proving it when \(d\ge g+r-1\). He use this theorem to construct new algebraic cohomology classes in \(\overline{\mathcal{M}}_{g,n}\) proving that \(\overline{\mathcal{M}}_{15,9}\) is of general type.