Isolated conics on a del Pezzo surface - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T13:31:14Z http://mathoverflow.net/feeds/question/63898 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/63898/isolated-conics-on-a-del-pezzo-surface Isolated conics on a del Pezzo surface mikhail skopenkov 2011-05-04T10:03:30Z 2011-05-11T09:52:43Z <p>Is there anything known about isolated conics in a del Pezzo surface: their number, arrangement, and the corresponding elements of the class group of surface's minimal desingularization? (Isolated means not belonging to a continuous family of conics in the surface.)</p> <p>A description similar to the one for isolated lines would be of most interest: "A del Pezzo surface has only finitely many lines. They correspond to curves E such that E^2 = E·K = −1 (so-called -1-curves) on the desingularization."</p> <p>More specifically, the question is about del Pezzo surfaces of degrees 5 and 6. References not requiring much background in algebraic geometry are greatly appreciated.</p> <p> And if we have a surface in C^3, whose linear normalization is a degree 5 or 6 Del Pezzo surface, can we say anything about isolated conics in this situation?</p> <p>[Edit2] I have found the following related result in the literature:</p> <p>"Any surface is a projection from its linear normalization. The projection is birational, and it preserves the degree of the surface and the degree of any curve not contained in the singular locus." </p> <p>Notice that the conics contained in the singular locus are also interesting for me.</p> <p>Additional question about surfaces in C^3 still unanswered.</p> http://mathoverflow.net/questions/63898/isolated-conics-on-a-del-pezzo-surface/63900#63900 Answer by J.C. Ottem for Isolated conics on a del Pezzo surface J.C. Ottem 2011-05-04T10:35:40Z 2011-05-04T10:51:32Z <p>While the number of lines on Del Pezzo surfaces are finite, the number of conics is infinite. More precisely, there are finitely many families $X\to P^1$ whose fibers are plane conics. Let me explain this in more detail.</p> <p>As you probably know, a degree $d$ Del Pezzo surface $X$ can be realized as the blow-up of $P^2$ in $r=9-d$ points in general position. The Picard group of $X$ has rank $r+1$ and is generated by the classes of the exceptional divisors $E_1,\ldots, E_r$ and $L$ which is the pullback of a general line in $P^2$ via the blow-up morphism $\pi:X\to P^2$. The intersection form on $N^1(X)=\mbox{Pic }X$ is given by $$E_i\cdot E_j=-\delta_{ij}, \qquad E_i\cdot L=1, \qquad L^2=1.$$Also, the anticanonical class equals $-K=3L-E_1-\ldots-E_r$ in this basis. </p> <p>If $X$ has degree $\ge 4$, then $-K$ is very ample, and the conics on $X$ correspond precisely to the effective divisor classes such that $$-K.D=2 \mbox{ and } D^2=0$$Examples are $L-E_i$ (pullback of a line through the point $p_i$) and $2L-E_1-E_2-E_3-E_4$ (pullback of a conic avoiding $p_5$). Using the AM-GM inequality, one can show that the number of such classes is finite.</p> <p>In fact it is easy to see that any conic can be written as the sum of two exceptional curves (which form the generators for the effective cone $\overline{NE}(X)$). So $D=E+F$ for some $E,F$ with $E.F=1$. Moreover, using this description, it is not hard to verify that the conic divisors $D$ are even base-point free and so by Riemann-Roch, define morphisms $X\to \mathbb{P}^1$. These morphisms are conic bundles, i.e., every fiber is isomorphic to a plane conic in $X$. </p> <p>On the other hand, the lines on $X$ correspond to classes satisfying $-K.E=1 \mbox{ and } E^2=-1$ so they don't 'move' in linear systems like the conics do, which explains why their number is finite.</p> <p>EDIT: There can not be any isolated conics on $X$, since if $D$ is any isolated rational curve, then $D^2&lt;0$ and the adjunction formula implies that $D^2=-1$, so $D$ is an exceptional curve, i.e., a line.</p> http://mathoverflow.net/questions/63898/isolated-conics-on-a-del-pezzo-surface/63901#63901 Answer by Dmitri for Isolated conics on a del Pezzo surface Dmitri 2011-05-04T10:36:18Z 2011-05-11T09:52:43Z <p>If by a del-Pezzo surface you mean what is written here : <a href="http://en.wikipedia.org/wiki/Del_Pezzo_surface" rel="nofollow">http://en.wikipedia.org/wiki/Del_Pezzo_surface</a>, i.e. surface such that $-K$ is ample, then there are no isolated conics on such surfaces at all. Indeed a smooth rational curve $C$ is isolated on a surface iff $C^2&lt;0$. On the other hand by adjunction formula we have $(K+C)C=-2$, i.e., $-KC= 2+C^2$. Hence $-KC$ is positive only on a rational curve $C$ only if $C^2\ge -1$. But if $C^2=-1$, then $C$ is a line, if $C^2\ge 0$ it is not isolated.</p> <p>Sometimes by del-Pezzo surface people mean rational surface with $-K$ semi-ample <em>and with $K^2>0$</em> (thanks to Artie for making this precise), as it is in the following article <a href="http://www.staff.science.uu.nl/~looij101/coble6.pdf" rel="nofollow">http://www.staff.science.uu.nl/~looij101/coble6.pdf</a> . More standard terminology for such surfaces are weak del-Pezzo surfaces. They indeed have exceptional curves $C$ with $C^2=-2$. The number of such curves is finite too. This is described, for example in the book of Dolgachev topics in classical algebraic geometry, beginning of chapter 8, <a href="http://www.math.lsa.umich.edu/~idolga/topics.pdf" rel="nofollow">http://www.math.lsa.umich.edu/~idolga/topics.pdf</a></p>