Are there (-2)-curves on an Enriques surface? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T06:05:19Z http://mathoverflow.net/feeds/question/52397 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/52397/are-there-2-curves-on-an-enriques-surface Are there (-2)-curves on an Enriques surface? fds 2011-01-18T10:52:27Z 2011-01-18T13:41:27Z <p>Let $X$ be an Enriques surface. A $(-2)$-curve is an irriducible rational curve on X such that $C^2 = -2$. By Proposition [VIII,16.1] from Barth-Peters-Van de Ven, we have that if $D^2 = -2$, then it is a $(-2)$-curve, but do such curves exist?</p> http://mathoverflow.net/questions/52397/are-there-2-curves-on-an-enriques-surface/52400#52400 Answer by EOP for Are there (-2)-curves on an Enriques surface? EOP 2011-01-18T11:25:06Z 2011-01-18T12:41:07Z <p>Since an Enriques surface is elliptic, it may have (-2)-curves as a component of singular fibers. You may refer S. Kondo, Enriques surfaces with finite automorphism groups. Japan. J. Math.(N.S.) 12 (1986), no. 2, 191--282. In the paper Kondo constructed explicitly many examples of Enriques surfaces with <em>finitely many</em> (-2)-curves.</p> http://mathoverflow.net/questions/52397/are-there-2-curves-on-an-enriques-surface/52401#52401 Answer by Francesco Polizzi for Are there (-2)-curves on an Enriques surface? Francesco Polizzi 2011-01-18T11:32:43Z 2011-01-18T13:15:36Z <p>As explained in J.C. Ottem's answer, the generic Enriques surface contains no smooth rational curves at all.</p> <p>However, it can happen that some special Enriques surface $X$ contains $(-2)$-curves, and also infinitely many of them (see <a href="http://www.springerlink.com/content/x175482343514234/" rel="nofollow">this paper</a> by Cossec and Dolgachev). The maximal number of <em>disjoint</em> $(-2)$ curves on $X$ is <em>eight</em>, and Enriques surfaces with eight disjoint $(-2)$-curves are classified in the article</p> <p>Mendes Lopes, Margarida; Pardini, Rita</p> <p>Enriques surfaces with eight nodes</p> <p>Math. Z. 241 (2002), no. 4, 673–683. </p> <p>The authors first show that, setting $C_1, \dots,C_8$ to be the exceptional $(-2)$-curves of $X$, the divisor $C_1+\dots+C_8$ is divisible by $2$ in the Picard group of $X$, or equivalently there exists a double cover $\widetilde{X} \to X$ branched exactly over them.</p> <p>The main theorem then states that an Enriques surface with eight disjoint $(-2)$-curves is isomorphic to $X=D_1\times D_2/G$, where $D_1,D_2$ are elliptic curves and $G$ is either $\mathbb{Z}_2^2$ or $\mathbb{Z}_2^3$.</p> http://mathoverflow.net/questions/52397/are-there-2-curves-on-an-enriques-surface/52402#52402 Answer by J.C. Ottem for Are there (-2)-curves on an Enriques surface? J.C. Ottem 2011-01-18T12:18:12Z 2011-01-18T13:41:27Z <p>It is well-known that (at least over $k=\mathbb{C}$) that the generic Enriques surface does not contain any smooth rational curves at all. This can be seen, for example, using the global Torelli theorem for Enriques surfaces. For a complete proof, see</p> <p>Barth, W., Peters, C.: Automorphisms of Enriques surfaces. Invent. Math. 73, 383-411 (1983).</p> <p>However, as Fransesco's answer shows, there are Enriques surfaces containing rational curves. Moreover, it is also known that once $S$ contains a rational curve, then generically it contains infinitely many. The reason for this is basically since the automorphism groups of Enriques surfaces tend to be very large.</p> <p>In fact, Cossec and Dolgatchev proved the following surprising result about rational curves on an Enriques surface:</p> <blockquote> <p>Let $S$ be an Enriques surface of degree $d$ in a projective space $\mathbb{P}^n$. Assume that S contains a smooth rational curve, then it contains such a curve of degree less or equal to $d$.</p> </blockquote> <p>This implies for example that the subset of the Hilbert scheme parametrizing Enriques surfaces of degree $d$ in $\mathbb{P}^n$ containing smooth rational curves is a constructible subset.</p>