Fractional indices of log Del Pezzo surfaces. (English. Russian original) Zbl 0724.14023

Alekseev, V. A.

Math. USSR, Izv. 33, No. 3, 613-629 (1989); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 52, No. 6, 1288-1304 (1988).

Let X be a normal projective complex surface having at most log-terminal singularities and such that the anti-canonical class \(-K_ X\) is ample (meaning that \(-mK_ X\) becomes an ample Cartier divisor for some integer \(m\geq 1)\). Such a surface will be referred to as a log Del Pezzo surface. For a log Del Pezzo surface X one can define the fractional index r(X) of X as the (positive) rational number \(r=r(X)\) such that \(- K_ X=rH\), where H is an ample Cartier divisor primitive in Pic(X). The paper under review describes the set \(R=\{r(X)| \quad X\quad a\quad \log Del Pezzo\quad surface\}\) by proving (among other things) the following theorem:
The set R has the following accumulation points: 0 and 1/m for any natural number m. All of these points are limit points from above and not from below. Moreover, for any natural number m, one can choose a sufficiently small punctured neighbourhood \(Q_ m=\{x\in {\mathbb{R}}| \quad 0<| x-1/m| <\epsilon_ m\}\) in such a way that all log Del Pezzo surfaces X with \(r(X)\in Q_ m\) can be classified explicitly.
[See also the correction of this paper announced below, ibid. 54, No.5, 1112 (1990).]

Reviewer: L.Bădescu (Bucureşti)

MSC:

14J25	Special surfaces
14J17	Singularities of surfaces or higher-dimensional varieties
14C20	Divisors, linear systems, invertible sheaves

Keywords:

ample anti-canonical class; log-terminal singularities; log Del Pezzo surface; fractional index