Log-canonical pairs and Gorenstein stable surfaces with

Abstract

We classify log-canonical pairs (X, ∆) of dimension two with KX +∆ an ample Cartier divisor with (KX + ∆) 2 = 1, giving some applications to stable surfaces with K 2 = 1. A rough classification is also given in the case ∆ = 0.

Computing invariants of semi-log-canonical surfaces

“…4.1 we prove by "classical" methods (restriction to a canonical curve) that if χ(X ) > 0 and K 2 = 1 then q(X ) = 0. By the results of [5], this implies that an irregular surface X with K 2 = 1 has χ(X ) = 0 and is a projective plane glued to itself along four lines. Secondly, in Sect.…”

Godeaux Surfaces with an Enriques Involution and Some Stable Degenerations

“…5.13] (cf. also [FPR14b,Thm. 3.2] for the Gorenstein condition) in order to construct a Gorenstein stable surface with K 2 = 1 with normalization (S,D), one has to give an involution ι of the normalization C φ * C ofD with the property that ι acts freely on the eight preimages of P 1 , .…”

“…We take ι to be the involution that exchanges C and φ * C and identifies C with φ * C via φ. One has χ(S) = 1 by [FPR14b,Prop. 3.4].…”

“…In §6 we give an explicit construction of the general Godeaux surface with an Enriques involution and use it to produce stable Godeaux surfaces. In this way we produce a normal Gorenstein degeneration with an elliptic singularity of degree 4, whose existence was predicted in [FPR14b], and we show the smoothability of one of the families of non-normal Godeaux surfaces with normalization isomorphic to P 2 ( [FPR14b], [FPR]). In addition we give examples of stable non-normal Godeaux surfaces with Cartier index equal to 2 whose normalization is not ruled, thus showing that the main result of [FPR14b] does not hold without the Gorenstein assumption.…”

“…§6 for the definition). The stable Gorenstein surfaces with K 2 = 1 (thus including the stable Gorenstein Godeaux surfaces) are investigated in the series of recent papers [FPR14b], [FPR14a], [FPR]. In §6 we give an explicit construction of the general Godeaux surface with an Enriques involution and use it to produce stable Godeaux surfaces.…”