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?
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3 Answers
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As explained in J.C. Ottem's answer, the generic Enriques surface contains no smooth rational curves at all.
However, it can happen that some special Enriques surface $X$ contains $(-2)$-curves, and also infinitely many of them (see this paper by Cossec and Dolgachev). The maximal number of disjoint $(-2)$ curves on $X$ is eight, and Enriques surfaces with eight disjoint $(-2)$-curves are classified in the article
Mendes Lopes, Margarida; Pardini, Rita
Enriques surfaces with eight nodes
Math. Z. 241 (2002), no. 4, 673–683.
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.
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 $\mathbb{Z}_2^3$.
answered Jan 18, 2011 at 11:32
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1$\begingroup$ Enriques surfaces always have Picard rank equal to 10, as follows from the Lefschetz (1,1) theorem. However, the effective cone is not constant in moduli which is what allows the number of (-2)-curves to vary. $\endgroup$– nafCommented Jan 18, 2011 at 12:16
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$\begingroup$ Yes, you are right. I added an answer below explaining this. $\endgroup$ Commented Jan 18, 2011 at 12:20
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$\begingroup$ You are definitely right. I corrected the answer, thank you $\endgroup$ Commented Jan 18, 2011 at 13:13
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$\begingroup$ The link to the paper by Cossec and Dolgachev at
springerlink.comis broken. I'm also unable to find any copy saved on the Wayback Machine. $\endgroup$ Commented Oct 1, 2022 at 9:42
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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
Barth, W., Peters, C.: Automorphisms of Enriques surfaces. Invent. Math. 73, 383-411 (1983).
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.
In fact, Cossec and Dolgatchev proved the following surprising result about rational curves on an Enriques surface:
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 $d$.
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.
answered Jan 18, 2011 at 12:18
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$\begingroup$ I think that the correct statement is: "once $X$ contains a smooth rational curve, then $generically$ it contains infinitely many of them". In fact, the general nodal Enriques surface has a big automorphism group, but some special Enriques surfaces have finite automorphism group (examples were given by Dolgachev, and are contained in the paper of Barth and Peters that you cited). $\endgroup$ Commented Jan 18, 2011 at 13:23
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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 finitely many (-2)-curves.
