
1. Computing higher codimension effective and nef cones in explicit examples

It is interesting to find explicit examples of varieties $X$ where it is possible to compute the effective cones of k-cycles.
1. Effective cones of projective bundles

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.

Problem 1.1.

What is the effective cone of surfaces $\text{Eff}_2(X)$ where $X$ is a $\mathbb{P}^2$-bundle over $\mathbb{P}^2$?
• Problem 1.2.

What is the effective cone of surfaces $\text{Eff}^2(X)$ where $X$ is a $\mathbb{P}^2$-bundle over a K3 surface?
• Problem 1.3.

If $\mathcal{E}\cong \mathcal{L}_1 \oplus ... \oplus \mathcal{L}_r$ is a split vector bundle on a smooth projective surface $S$, what are the effective cones of k-cycles $\text{Eff}_k(\mathbb{P}(\mathcal{E}))$?
• Effective cones of hypersurfaces

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.

Problem 1.4.

If $X\subset Gr(m,n)$ is a very general hypersurface, what is the effective cone of $k$-cycles $\text{Eff}_k(K)$?
• Problem 1.5.

If $X\subset \mathbb{P}^n\times \mathbb{P}^m$ is a very general hypersurface, what is the effective cone of $k$-cycles $\text{Eff}_k(X)$?
•     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.

Problem 1.6.

If $X\subset Y$ is a very general ample divisor of sufficiently high degree, does the pushforward map $i_*: N_k(X) \rightarrow N_k(Y)$ induce an isomorphism of effective cones for $k$ sufficiently small?
• Effective cones of Hilbert schemes and Moduli Spaces

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.

Problem 1.7.

What is the effective cone of surfaces $\text{Eff}_2(S^{[n]})$? What is the codimension 2 effective cone $\text{Eff}^2(S^{[n]})$? What about higher dimension/codimension?
•     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 $\overline{\mathcal{M}}_{g,n}^{\mathfrak{S}_n}:=\overline{\mathcal{M}}_{g,n}/{\mathfrak{S}_n}$.

Problem 1.8.

What is the effective cone of surfaces $\text{Eff}_2(\overline{\mathcal{M}}_{0,7}^{\mathfrak{S}_7})$?
• Effective cones of blowups

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 $\mathbb{P}^n$.

Problem 1.9.

Let $Z\subset \mathbb{P}^n$ be a smooth subvariety with $1\le \text{dim}Z \le n-2$. What is the effective cone of $k$-cycles $\text{Eff}_k(X)$?
1. Remark. In question 9 it is natural to first consider a few basic cases such as the blowup of a union of a small number of lines in $\mathbb{P}^n$ or the blowup of low degree, low codimension complete intersections.

Cite this as: AimPL: Positivity of cycles, available at http://aimpl.org/poscycles.