Complements to square-zero curves in projective surfaces sometimes have several non-isomorphic algebraic structures. Serre’s example is possibly the most famous illustration of this phenomenon (see f.e. [PS,Ex. 4.19]). Here is another example.

Consider a pencil of cubic curves in $\mathbb P^2$. They pass through 9 common points. The blow up $X$ of $\mathbb P^2$ in these points admits an elliptic fibration $f\colon X\to \mathbb P^1$. Let $D$ be a smooth fiber of this fibration, or a fiber of type $I_m, m\ge 1$. One can define a logarithmic transform $X’$ of $X$ in $D$, which is an elliptic surface biholomorphic to $X$ outside of $D$ but having a fiber $D’$ of multiplicity $n$ instead of $D$ ([BHPV, Ch. V.13]).

The construction is purely complex analytic, yet, in our case $X’$ happens to have an algebraic structure. The first way to see this is to use results of [FM] (I’ll put the precise reference tomorrow, sorry). They prove that a logarithmic transform of an elliptic surface with singular fibers is always projective up to a Shafarevich—Tate twist. The Shafarevich—Tate group of a rational elliptic surface is trivial, hence the claim. Also, one can notice that the elliptic surface $X’$ is a blow up of 9 points $\{p_1,..p_9\}$ on an elliptic curve $E$ in $\mathbb P^2$. The points must satisfy the condition that $\mathcal O_E(p_1+…+p_9)\otimes \mathcal O_{\mathbb P^2}(-3)|_E$ is an $n$-torsion line bundle.

The complements $X-D$ and $X’-D’$ are biholomorphic but the algebraic structures induced from $X$ and $X’$ respectively are not preserved under this isomorphism. In this fashion we construct countable number of algebraic structures on $X-D$.

Question 1: Let us blow up 9 points on $\mathbb P^2$ in a general position, so that there is only one elliptic curve passing through them. Let $E$ be the strict transform of this elliptic curve. Does the complement to $E$ have several algebraic structures?

And here is a more general question.

Question 2: Do you know other examples of non-isomorphic algebraic structures on complements to square-zero curves?

Answers to any of them will be greatly appreciated!

Bibliography:

1. [PS] Peters, Steenbrink. Mixed Hodge structures.
2. [BHPV] Barth, Hulek, Peters, Van de Ven. Compact complex surfaces.
3. [FM] Friedman, Morgan. Smooth four-manifolds and complex surfaces.