Non-rigid quartic 3-folds

Let X C P4 be a terminal factorial quartic 3-fold. If X is non-singular, X is birationally rigid, i.e. the classical minimal model program on any terminal Q-factorial projective variety Z birational to X always terminates with X. This no longer holds when X is singular, but very few examples of non-rigid factorial quartics are known. In this article, we first bound the local analytic type of singularities that may occur on a terminal factorial quartic hypersurface X c P4. A singular point on such a hypersurface is either of type cAn (n > or equal 1), or of type cDm (m> or equal 4), or of type cE6, cE7 or cE8. We first show that if (P e X) is of type cAn, n is at most 7, and if (P \in X) is of type cDm, m is at most 8. We then construct examples of non-rigid factorial quartic hypersurfaces whose singular loci consist (a) of a single point of type cAn for 2\leq n\leq 7 (b) of a single point of type cDm for m= 4 or 5 and (c) of a single point of type cEk for k=6,7 or 8

The Sarkisov program for Mori fibred Calabi-Yau pairs

We prove a version of the Sarkisov program for volume-preserving birational maps of Mori fibred Calabi-Yau pairs valid in all dimensions. Our theorem generalises the theorem of Usnich and Blanc on factorisations of birational maps of the 2-dimensional torus that preserve the volume form dx/x ^ dy/y

On toric geometry and K-stability of Fano varieties

We present some applications of the deformation theory of toric Fano varieties to K-(semi/poly)stability of Fano varieties. First, we present two examples of K-polystable toric Fano $3$-fold with obstructed deformations. In one case, the K-moduli spaces and stacks are reducible near the closed point associated to the toric Fano $3$-fold, while in the other they are non-reduced near the closed point associated to the toric Fano $3$-fold. Second, we use openness of K-semistability to show that the general members of two deformation families of smooth Fano $3$-folds are K-semistable by building degenerations to K-polystable toric Fano $3$-folds.Comment: 27 page

Finite generation and geography of models

There are two main examples where a version of the Minimal Model Program can, at least conjecturally, be performed successfully: the first is the classical MMP associated to the canonical divisor, and the other is Mori Dream Spaces. In this paper we formulate a framework which generalises both of these examples. Starting from divisorial rings which are finitely generated, we determine precisely when we can run the MMP, and we show why finite generation alone is not sufficient to make the MMP work.Engineering and Physical Sciences Research Council [grant number EP/H028811/1]; DFG-Forschergruppe 790 \Classi cation of Algebraic Surfaces and Compact Complex Manifolds", and by the OTKA Grants 77476 and 81203 of the Hungarian Academy of Sciences

The Calabi problem for smooth Fano threefolds

To be published by CUP, LMS Lecture Notes Series 2022Copyright © 2021 The Authors. There are 105 irreducible families of smooth Fano threefolds, which have been classified by Iskovskikh, Mori and Mukai. For each family, we determine whether its general member admits a K¨ahler–Einstein metric or not. We also find all K¨ahler–Einstein smooth Fano threefolds that have infinite automorphism groups.Engineering & Physical Sciences Research Council (EP/056689/1 Calabi conjecture for smooth Fano threefolds); Heilbronn Institute for Mathematical Research (K-stability of smooth Fano 3-folds).https://archive.mpim-bonn.mpg.de/id/eprint/4589/1/mpim-preprint_2021-31.pd

Compactifications of spaces of Landau-Ginzburg models

This paper reviews results and techniques from the authors' previous work "Symplectomorphism group relations and degenerations of Landau-Ginzburg models" and applies them in basic examples. The main example is the $A_n$ category where we observe a relationship to stability conditions and directed quiver representations. We conclude with a brief survey of applications to the birational geometry of del Pezzo surfaces