Positivity of cycles

    In [MR2836078] Fulger showed how to compute the effective cone of $k$-cycles $\text{Eff}^k(\mathbb{P}(\mathcal{E})$ of a projective bundle over a smooth curve. In this case the answer can be given in terms of data coming from the Harder-Narasimhan filtration of $\mathcal{E}$. However, even in the case where $X$ is a $\mathbb{P}^2$-bundles over a surface $S$, it is not known how to compute $\text{Eff}_2(X)$. It is natural to first consider the case where $S$ is the projective plane or a K3 surface.

    For very basic varieties such as $Gr(m,n)$ or $\mathbb{P}^n\times \mathbb{P}^m$ the effective cones of $k$-cycles can be computed. It then natural to ask if it is possible to compute the effective cones of very general hypersurfaces in these varieties.

    In the above cases, assuming $i:X\hookrightarrow \mathbb{P}^n\times\mathbb{P}^m$ or $i:X\hookrightarrow Gr(k,n)$ is ample then the Lefschetz hyperplane theorems imply that $i_*:N_k(X) \rightarrow N_k(Gr(k,n))$ and $i_*:N_k(X) \rightarrow N_k(\mathbb{P}^n\times\mathbb{P}^m)$ are isomorphisms for $k<\text{dim}(X)/2$. It is natural to ask if there is any Lefschetz type phenomena for effective cones of very general hypersurfaces of sufficiently high degree.

    It is natural to try to compute effective cones in the case of Hilbert schemes or moduli spaces. One natural Hilbert scheme to consider first is the Hilbert scheme of length $n$ subscheme of a smooth surface $S$, denoted by $S^{[n]}$. A great deal is known about these Hilbert schemes. In recent years there has been a lot of exciting work on the effective cones of divisors and curves of the Hilbert schemes $\text{Eff}^1(S^{[n]})$ and $\text{Eff}_1(S^{[n]})$. See for example Bayer and Macri’s work on Hilbert schemes of K3 surfaces [MR3279532] or Huizenga’s work on Hilbert schemes of $\mathbb{P}^2$ [MR3419956]. It is then natural to ask how to compute the effective cone of surfaces or the codimension 2 effective cones.

    Historically there has also been fervent interest in computing the effective cones of divisors and curves in the case of the moduli space of genus $g$ curves $\overline{\mathcal{M}}_g$ or their pointed versions $\overline{\mathcal{M}}_{g,n}$. In recent years for example there has been a great deal of work on the so called F-conjecture which predicts the effective cone of curves in $\overline{\mathcal{M}}_{0,n}$ is generated by explicit curve classes (so called "F-curves"), see e.g. Gibney’s work [MR2551995]. It is then natural to ask about higher dimensional effective cones for small values of $g$ and $n$. In the pointed case, to make the problem even simpler one can ask about the symmetrized space $\overline{\mathcal{M}}_{g,n}^{\mathfrak{S}_n}:=\overline{\mathcal{M}}_{g,n}/{\mathfrak{S}_n}$.

    Lastly, it is natural to ask what the relationship is between effective cones of a variety and its blowup. This is a deceptively difficult question already in the case of the effective cone of curves and the blowup of 10 points in $\mathbb{P}^2$, which is the setting of Nagata’s conjecture. However, the question of computing effective cones of blowups is quite possibly more tractable when the center of the blowup is a positive dimensional subvariety of $\mathbb{P}^n$.