Gorenstein stable surfaces with KX2=1 and pg>0

Abstract

In this paper we consider Gorenstein stable surfaces with KX2=1 and positive geometric genus. Extending classical results, we show that such surfaces admit a simple description as weighted complete intersection. We exhibit a wealth of surfaces of all possible Kodaira dimensions that occur as normalisations of Gorenstein stable surfaces with KX2=1; for pg=2 this leads to a rough stratification of the moduli space. Explicit non‐Gorenstein examples show that we need further techniques to understand all possible degenerations.

Gorenstein stable surfaces with $K_X^2 = 2$ and $\chi(\mathcal O_X) = 4$

“…Let k be the number of elliptic singularities of X and suppose that the minimal resolution Y of X satisfies κ(Y ) = −∞. Then either Y is rational with χ (O Y ) = 1 and k = 3, or χ (O Y ) = 0 and k = 4; in the latter case, the four elliptic singularities are simple.Proof after Franciosi, Pardini and Rollenske[16]. With the same proof as in Franciosi, Pardini, Rollenske[18, Lemma 4.5] we get that either Y is rational ( χ (O Y ) = 1), or Y min is ruled of genus 1 ( χ (O Y ) = 0) and that in the latter case all elliptic singularities are simple.…”

“…is the blow up in one point and X has a unique elliptic singularity of degree Proof after Franciosi, Pardini and Rollenske [16]. Let E i ⊂ Y , i = 1, .…”

“…The following two results are part of a manuscript in preparation by Franciosi, Pardini and Rollenske [16]. For simplicity, we restrict both to the relevant case.…”

“…For the locus of non-normal surfaces, however, this is much more complicated, as we will indicate in Example 4.18 and Example 4. 19. This is why on the locus of non-normal surfaces, the stratification will not be fine enough to control the Hodge type.…”

Optimal bounds for T-singularities in stable surfaces

“…We realize one of the five cases, which gives construction of normal stable surfaces W with one singularity 1 25 (1, 9), p g (W ) = 2, q(W ) = 0, and K 2 W = 1. There is a recent study of stable surfaces for those invariants in [FPR17], and this example seems to be new. The surface W has obstructions, and so we do not know if it is Q-Gorenstein smoothable.…”