Curves with prescribed intersection with boundary divisors in moduli spaces of curves (Q2629763)

Let \(\tilde{\mathcal M}_{0,n}\) be the moduli space of stable genus zero curves with \(n\) unordered markings. Consider the set \(\mathcal B\) of irreducible boundary divisors of \(\tilde{\mathcal M}_{0,n}\). In this paper, the author studies the following question: given any nonempty subset \(\mathcal B'\) of \(\mathcal B\), does there exist a curve \(C\) in \(\tilde{\mathcal M}_{0,n}\) such that the boundary divisors that intersect \(C\) are precisely those in \(\mathcal B'\)? The main result gives an affirmative answer for \(n \leq 17\). For general \(n\), the author is able to answer the question for a number of special collections of boundary divisors. As an application, the author answers the same type of questions for the moduli space of stable curves and the moduli space of hyperelliptic curves in low genus.