Let $S$ be an Enriques surface, i.e. a quotient of a K3 surface by a free involution. Enriques surfaces arise as elliptic fibrations $S\rightarrow \mathbb{P}^1$ with 12 singular fibers and 2 double points.

Claim There exist elliptic surfaces $S_1, S_2$ such that we can degenerate the elliptic fibration $S\rightarrow \mathbb{P}^1$ to get $$ S_1 \cup_E S_2 \rightarrow \mathbb{P}^1 $$ wherethe intersection $E$ is a common elliptic fiber.

Question How can I prove this claim?

More precisely, the elliptic surfaces $S_1, S_2$ are obtained as follows:

Let $E\times \mathbb{P}^1$ be the product of elliptic curve and a projective line. Let $t$ denote the translation by a 2-torsion point on $E$, and let $\iota$ be an involution of $\mathbb{P}^1$. Then $(t_2,\iota)$ is a fixed point free involution on $E\times \mathbb{P}^1$, and we define $$ S_1:=(E\times \mathbb{P}^1)/\langle (t_2,\iota) \rangle. $$ By projecting right, $S_1\rightarrow \mathbb{P}^1/\langle \iota \rangle$ is an elliptic fibration with no singular fibers and 2 double fibers.

Let $S_2$ be the blow-up of $\mathbb{P}^2$ in 9 points. $S_2$ admits an elliptic fibration $S_2\rightarrow \mathbb{P}^1$ with 12 singular fibers and no double fibers.


1 Answer 1


The surface $S_1$ is rational and the surface is an elliptic ruled surface (induced by the projection $E\times \mathbb{P}^1\to E$). The glued surface is a standard Type II degeneration of Enriques surfaces (an elliptic ruled chain) classified by Vic. Kulikov and D. Morrison. I believe that the general member of the deformation is the Enriques surface obtained by the logarithmic transformation at two isomorphic nonsingular fibers corresponding to the choice of the same 2-torsion on each fiber. But this has to be checked.


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