References about pseudoeffective cone - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T06:06:10Z http://mathoverflow.net/feeds/question/78115 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/78115/references-about-pseudoeffective-cone References about pseudoeffective cone fds 2011-10-14T09:59:10Z 2011-10-14T11:28:43Z <p>I'm looking for references of explicit computation of the pseudoeffective cone $\overline{\text{Eff}}(X)$ of a projective variety $X$.</p> http://mathoverflow.net/questions/78115/references-about-pseudoeffective-cone/78117#78117 Answer by J.C. Ottem for References about pseudoeffective cone J.C. Ottem 2011-10-14T10:35:52Z 2011-10-14T11:28:43Z <p>I'd say Lazarsfeld's book "<a href="http://books.google.com/books/about/Positivity_in_Algebraic_Geometry_I.html?id=T87ftUcU_hEC" rel="nofollow">Positivity in algebraic geometry I,II</a>" is the standard reference these days. In particular, Volume I has a lot of explicit examples. I also recommend Debarre's book <a href="http://books.google.com/books/about/Higher_dimensional_algebraic_geometry.html?id=Mtm7Zi-7U1gC" rel="nofollow">'Higher dimensional algebraic geometry"</a> which is similar in style with a lot of nice examples and explicit computations.</p> <p>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 <a href="http://www.cs.amherst.edu/~dac/toric.html" rel="nofollow">'Toric varieties'</a> for details. This gives a hoard of interesting examples.</p> <p>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:</p> <p><strong>Example</strong>. 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 $\tau\subset \overline{Eff}(X)$. Coversely, 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 $\tau^*$.</p> <p>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. </p> <p>For material on the effective cones of surfaces, see for example </p> <p>B. Harbourne <a href="http://arxiv.org/abs/0907.4151" rel="nofollow">"Global aspects of the geometry of surfaces"</a> and </p> <p>Y. Tschikel "<a href="http://www.cims.nyu.edu/~tschinke/papers/yuri/08cmi/cmi4.pdf" rel="nofollow">Algebraic varieties with many rational points</a>. </p> <p>For K3 surfaces, S. Kovacs has a nice paper on the <a href="http://www.springerlink.com/content/r654368027w422r2/" rel="nofollow">'Cone of curves of a K3 surface'</a> (see also <a href="http://mathoverflow.net/questions/42699" rel="nofollow">this answer</a>). There are also many explicit examples in Artebani-Hausen-Laface's paper <a href="http://arxiv.org/abs/0901.0369" rel="nofollow">On Cox rings of K3-surfaces</a>. </p> <p>I can also recommend Artie Prendergast-Smith's papers at <a href="http://www.math.uic.edu/~artie/" rel="nofollow">his homepage</a>. In particular, his PhD thesis contains a very explicit example where he computes the of the effective cone of a rational threefold.</p> <p>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:</p> <p>A. Laface, M. Velasco, <a href="http://arxiv.org/abs/0810.3730" rel="nofollow">A survey on Cox rings</a></p> <p>I. Arzhantsev, U. Derenthal, J. Hausen, A. Laface, <a href="http://arxiv.org/abs/1003.4229" rel="nofollow">Cox rings</a></p> <p>J. Gonzalez, M. Hering, S. Payne, H. Süß <a href="http://arxiv.org/abs/1009.5238" rel="nofollow">Cox rings and pseudoeffective cones of projectivized toric vector bundles</a> and</p> <p>M. Artebani, A. Laface <a href="http://arxiv.org/abs/0909.1835" rel="nofollow">Cox rings of surfaces and the anticanonical Iitaka dimension</a></p>