Weierstrass meets Enriques

We study in detail the degeneration of K3 to T-4/Z(2). We obtain an explicit embedding of the lattice of collapsed cycles of T-4/Z(2) into the lattice of integral cycles of K3 in two different ways. Our first method exploits the duality to the heterotic string on T-3. This allows us to describe the degeneration in terms of Wilson lines. Our second method is based on the blow-up of T-4/Z(2). From this blow-up, we directly construct the full lattice of integral cycles of K3. Finally, we use our results to describe the action of the Enriques involution on elliptic K3 surfaces, finding that a Weierstrass model description is consistent with the Enriques involution only in the F-theory limit.