References about pseudoeffective cone

Question

I'm looking for references of explicit computation of the pseudoeffective cone $\overline{\text{Eff}}(X)$ of a projective variety $X$.

Answer 1

I'd say Lazarsfeld's book "Positivity in algebraic geometry I,II" is the standard reference these days. In particular, Volume I has a lot of explicit examples. I also recommend Debarre's book 'Higher dimensional algebraic geometry" which is similar in style with a lot of nice examples and explicit computations.

If you are familiar with toric geometry, there is (not surprisingly) a simple description of the psedudoeffective cone in terms of the combinatorial data in the fan. See Cox-Little-Schenck's new book 'Toric varieties' for details. This gives a hoard of interesting examples.

If you are looking for examples with non-toric varieties, I'd recommend starting with the case where $X$ is a surface. In that case the effective cone coincides with the cone of curves and can be studied using the intersection form. Let me give an example:

Example. Let $X$ be the blow-up of $\mathbb{P}^2$ at two points and let $E_1,E_2$ be the exceptional divisors. A basis for $Pic(X)$ is given by $L,E_1,E_2$ where $L$ is the pull back of a general line in $\mathbb{P}^2$. We show that $\overline{Eff}(X)$ is spanned by $E_1,E_2$ and the strict transform of the line $L_0=L-E_1-E_2$. Let $\tau$ be the cone spanned by these three classes. Since they are all effective we have $\tau\subset \overline{Eff}(X)$. Conversely, let $D$ be any effective divisor with class $aL+bE_1+cE_2$. We will show that $D$ can be written as a sum of elements from $\tau$. We may assume $D$ to be irreducible. If $D$ is not one of the $E_1,E_2,L_0$, we then have $D.E_i\ge 0$ and $D.L_0\ge 0$. In particular, $D$ belongs to the dual cone of $\tau$, which is easily computed as $\tau^*=\langle L,L-E_1,L-E_2\rangle_{\ge 0}$. Now $L, L-E_1, L-E_2$ are all effective and can be written as positive linear combinations of $E_1,E_2,L_0$, and hence so can $D$. As a by-product, we have just computed the nef cone, which is $\tau^*$.

Of course, this example is in fact toric, but the main point is that this type of argument works for more general surfaces, as long as you have a good description of the surface. For example, the argument above generalizes to show that a Del Pezzo surface, $\overline{Eff}(X)$ is spanned by the $(-1)$-curves on $X$ (this is shown in Debarre's book, I think). In general, the effective cone of rational surfaces have been studied a lot using their models as blow-ups.

For material on the effective cones of surfaces, see for example

B. Harbourne "Global aspects of the geometry of surfaces" and

Y. Tschikel "Algebraic varieties with many rational points.

For K3 surfaces, S. Kovacs has a nice paper on the 'Cone of curves of a K3 surface' (see also this answer). There are also many explicit examples in Artebani-Hausen-Laface's paper On Cox rings of K3-surfaces.

I can also recommend Artie Prendergast-Smith's papers at his homepage. In particular, his PhD thesis contains a very explicit example where he computes the of the effective cone of a rational threefold.

In addition to pseduoeffective cones, you might also be interested in seeing explicit computations of Cox rings, which are graded by the monoid of effective divisors (in particular if you have a description of the Cox ring, you know all about the effective cone). Here I can recommend the following papers:

A. Laface, M. Velasco, A survey on Cox rings

I. Arzhantsev, U. Derenthal, J. Hausen, A. Laface, Cox rings

J. Gonzalez, M. Hering, S. Payne, H. Süß Cox rings and pseudoeffective cones of projectivized toric vector bundles and

M. Artebani, A. Laface Cox rings of surfaces and the anticanonical Iitaka dimension