Normal Gorenstein Surfaces with Ample Anti-canonical Divisor

Characterization of projective spaces by Seshadri constants

“…We claim that if S is a Gorenstein log del Pezzo surface of degree d ≥ 3, then there exists an irreducible curve C ∈ | − K S | with a double point p lying in the smooth locus of S. After blowing up d − 3 general points on S, it suffices to prove the claim when d = 3, in which case S is a nodal cubic surface in P 3 by [10,Theorem 4.4]. But then there are only finitely many lines on S whereas by dimension count there exists C ∈ | − K X | that is singular at any given p ∈ S, hence the claim follows immediately.…”

Classification of log del Pezzo surfaces of index three

“…Log del Pezzo surfaces with small index have been studied by many authors. The classification of log del Pezzo surfaces with index one (that is, with at most rational double points) is well-known (see [Bre80], [Dem80], [HW81]). …”

On Gorenstein surface singularities with fundamental genuspf≧ 2 which satisfy some minimality conditions

“…If (X, x) is a Gorenstein surface singularity, the following inequality holds (cf. [5,14,20] for general case and [24,25] for case of p g = 2):…”