\(k\)-canonical divisors through Brill-Noether special points (Q6609491)

The vector bundle of stable \(k\)-differentials over \(\overline{\mathcal{M}}_g\), the moduli space of stable curves of genus \(g\), is called \(\mathbb{E}^k_g\). Its projectivization \(\mathbb{PE}^k_g\) compactifies the moduli space of \(k\)-canonical divisors on smooth algebraic curves. A number of divisor classes on \(\mathbb{PE}^k_g\) have been studied. In the present paper the authors consider divisors consisting of \(k\)-differentials vanishing at a Brill-Noether special point.\N\NIn Theorem 1.1, the class of the closure of such a divisor is computed. Two independent proofs of this result are given in Sections 3 and 4, respectively. The first one involves the incidence divisor \(\mathbb{H}^k_{g,1}\). In the second one, the authors consider the locus \(\mathbb{H}^{\boldsymbol{a}}_{g,d}\) and intersect it with some families of curves, including curves on K3 surfaces, DuVal curves, and hyperelliptic curves.\N\NSection 5 is devoted to prove Theorem 1.4, according to which the class of \(\mathbb{H}^k_{g,1}\) is rigid and extremal for \(k \in \{1,2\}\). The question on whether this result is true for \(k \geq 3\) is posed, and its study has been started by the authors.