Two local inequalities

Del Pezzo Surfaces and Local Inequalities

“…I can not use Theorem 13 instead of Theorem 5 in general. I checked this in many cases considered in [2]. To convince the reader, let me give Example 16.…”

“…By [7,Excercise 6.26], an analogue of Theorem 5 holds for surfaces with at most quitient singularities. This helped me to find in [2] many new applications of Theorem 5 that do not follow from Theorem 4. Remark 6.…”

“…Then I try to bound a 1 and a 2 using some global methods. Usually, I end up with two non-negative rational numbers α and β such that αa 1 + βa 2 1. Put M = D · ∆ 1 , N = C · ∆ 2 , A = −∆ 2 1 , and B = −∆ 2 1 .…”

Equivariant Burnside groups and representation theory

“…For example, there are two actions of A 5 on P 2 , coming from the two nonisomorphic 3dimensional represesentations of A 5 ; they are not conjugated in PGL 3 but are conjugated in the Cremona group Cr 2 [4, Remark 6.3.9]. On the other hand, A 6 and PSL 2 (F 7 ) also admit two actions on P 2 , but these are not conjugated in Cr 2 [2,Theorems B.8 and A.20].…”

On Conjugacy Classes of the Klein Simple Group in Cremona Group

“…This double cover is known as the del Pezzo surface of degree 2, which is the blow up of P 2 at 7 points in general position (denoted by dP 2 ). Both these varieties turn out to be PSL 2 (F 7 )-birationally rigid, in particular P 2 is not PSL 2 (F 7 )-equivariantly birational to dP 2 , hence the two embeddings are non-conjugate (see [5] and Appendix B therein). Section 2 reviews the stable non-conjugacy of PSL 2 (F 7 ) in Cr 2 (C) based on the work of Prokhorov and Bogomolov.…”