Are there (-2)-curves on an Enriques surface? - MathOverflow

Are there (-2)-curves on an Enriques surface?

2011-01-18T10:52:27Z

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?

Answer by EOP for Are there (-2)-curves on an Enriques surface?

2011-01-18T11:25:06Z

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.

Answer by Francesco Polizzi for Are there (-2)-curves on an Enriques surface?

2011-01-18T11:32:43Z

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 $\mathbb{Z}_2^3$.

Answer by J.C. Ottem for Are there (-2)-curves on an Enriques surface?

2011-01-18T12:18:12Z

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.