Alpha invariant for general polarizations of del Pezzo surfaces of degree 1

Abstract

For an arbitrary ample divisor A in smooth del Pezzo surface S of degree 1, we verify the condition of the polarization (S,A) to be K‐stable and it is a simple numerical condition.

On a Conjecture of Hong and Won

“…Hong and Won suggested an answer to Problem 1.5 for del Pezzo surfaces of degree one. This answer is given by their [9,Conjecture 4.3], which is Conjecture 2.1 in Section 2.…”

“…For smooth del Pezzo surfaces, Theorem 1.3 gives Theorem 1.4 ( [9,2]). Let S be a smooth del Pezzo surface such that K 2 S = 1 or K 2 S = 2.…”

“…Let us describe the structure of this paper. In Section 2, we describe[9, Conjecture 4.3]. In Section 3, we present several well known local results about singularities of log pairs.…”

Stable Polarized del Pezzo Surfaces

“…For smooth del Pezzo surfaces of degree one, the assertion of Theorem 1.1 was recently proved by Hong and Won in [13] using a different approach.…”