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Suppose $X$ is a surface, can it have infinitely many $(-1)$ curves?(If they are disjoint, we can see this since Neron Severi group has finite rank, but how to deal with the case when they are not disjoint?)

Suppose $X$ is a projective variety, is it true that the codimension $1$ subvarieties with negative top self-intersection (degree of the zero cycle) is finite?

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Let $C_1$ and $C_2$ be two general cubic plane curves. Let $S$ be the surface obtained by blwoing-up the nine base points. Then $S$ is a rational elliptic surface. Every section of the elliptic fibration $S\to \mathbb{P}^1$ is a $-1$ curve. After choosing one section as the zero-section you find that the sections form an abelian group. If $C_1$ and $C_2$ are sufficiently general then this group is isomorphic with $\mathbb{Z}^8$, in particular you have infinitely many $-1$ curves.

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  • $\begingroup$ Is the abelian group of sections finitely generated? How many disjoint sections it has at most？ $\endgroup$
    – user39380
    Commented Sep 23, 2015 at 13:05
  • $\begingroup$ The Picard group of $S$ has rank 10, hence there are at most 9 disjoint -1 curves. If $C_1$ and $C_2$ intersect in 9 distinct points then there are 9 disjoint -1 curves. For rational elliptic surfaces the group of sections is finitely generated. $\endgroup$
    – Remke Kloosterman
    Commented Sep 23, 2015 at 13:25
  • $\begingroup$ If we project a section to $\mathbb{P}^2$, we will get a line, but any line would meet a cubic in $3$ points, but the section should meet a fiber in one point, I wasn't quite understanding this.. $\endgroup$
    – user39380
    Commented Sep 23, 2015 at 14:05
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    $\begingroup$ There are many type of sections. First the 9 exceptional divisors are sections. Then the lines through 2 of the 9 base points are sections, conics through 5 of the 9 base points are sections, cubics through 7 of the 9 base points, which have a node at one of the 7 points are also sections, etc. $\endgroup$
    – Remke Kloosterman
    Commented Sep 23, 2015 at 17:43
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More generally, the surface obtained by blowing up $r\geq 9$ general points in $\mathbb{P}^2$ contains infinitely many $(-1)$ curves. See Nagata, On rational surfaces, II, Mem. College Sci. Univ. Kyoto Ser. A Math. Volume 33, no. 2 (1960), 271-293, where the term "general" is explained in a precise way.

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