August 2022, Pohang

Title:Families of simple subgroups in the Cremona group arising from del Pezzo fibrations

Abstract:
Cremona group of rank n is the group of birational self-maps of the projective space of dimension n. For any subgroup G of Cremona group there is a rational variety on which G acts regularly. This allows to translate the study of subgroups of Cremona group into the study of G-equivariant geometry of rational varieties. In this talk I will describe some continuous families of rational threefolds with an action of alternating group of rank 5. I will also explain why the corresponding subgroups of the Cremona group are not pair-wise conjugate.

May 2022

2n^2-inequality for cA_1 points and applications to birational rigidity
with Takuzo Okada, Erik Paemurru, and Jihun Park

Abstract: The 4n^2-inequality for smooth points plays an important role in the proofs of birational (super)rigidity. The main aim of this paper is to generalize such an inequality to terminal singular points of type cA_1, and obtain a 2n^2-inequality for cA_1 points. As applications, we prove birational (super)rigidity of sextic double solids, many other prime Fano 3-fold weighted complete intersections, and del Pezzo fibrations of degree 1 over P1 satisfying the K^2-condition, all of which have at most terminal cA1 singularities and terminal quotient singularities. These give first examples of birationally (super)rigid Fano 3-folds and del Pezzo fibrations admitting a cA1 point which is not an ordinary double point.