Talks
All times are in GMT (the current time in the UK). All talks are hosted on Microsoft Teams. The links in the abstracts below are to the corresponding Microsoft Teams event for that talk. See the Registration page for further details about joining the talks.
For a list of talks ordered by date and time, see the Schedule page.
Hamid Ahmadinezhad (Loughborough) - Seshadri constants, induction, and K-stability
25 February, 11:30-12:30
I will talk about an inductive approach to proving K-stability of Fano varieties. As a tool, I introduce a new bound on the \delta-invariant using Seshadri constants and conclude several K-stability results. This is join work with Ziquan Zhuang.
Gavin Brown (Warwick) - Some normal forms for flops
23 February, 11:30-12:30
I describe ongoing work with Michael Wemyss to understand crepant contractions from smooth 3-folds, in particular simple flopping contractions, via their noncommutative contraction algebras and the noncommutative potentials that govern them.
Cinzia Casagrande (Torino) - On Fano 4-folds with Lefschetz defect 3
24 February, 11:30-12:30
We will talk about a classification result for some (smooth, complex) Fano 4-folds. We recall that if X is a Fano 4-fold, the Lefschetz defect delta(X) is an invariant of X defined as follows. Consider a prime divisor D in X and the restriction r: H^2(X,R)->H^2(D,R). Then delta(X) is the maximal dimension of ker(r), where D varies among all prime divisors in X. In a previous work, we showed that if X is not a product of surfaces, then delta(X) is at most 3, and if moreover delta(X)=3, then X has Picard number 5 or 6. We will explain that in the case where X has Picard number 5, there are 6 possible families for X, among which 4 are toric. This is a joint work with Eleonora Romano.
Daniel Cavey (Nottingham) - Restrictions on the Singularity Content of a Fano Polygon
26 February, 15:30-16:30
Singularity content is a combinatorial property of a Fano polygon that describes geometric properties of the qG-smoothing of the corresponding toric Fano variety. We determine restrictions on the singularity content to derive geometric results for certain orbifold del Pezzo surfaces.
Ivan Cheltsov (Edinburgh) - K-stability of smooth Fano threefolds
23 February, 14:00-15:00
I will explain which smooth Fano threefolds are K-stable and K-polystable.
Tom Ducat (Durham) - Reid’s pagoda (and other non-toric flops) done 'torically'
24 February, 16:00-17:00
Everyone knows that the Atiyah flop can be described in terms of toric geometry by subdividing a square cone in two different ways. Reid’s pagoda is a geometric construction giving the flop of (-2,0)-curve and, as such, can’t be described in terms of toric geometry. Nevertheless, I will explain how to obtain the pagoda by subdividing a cone in an integral affine manifold in two different ways.
Andrea Fanelli (Bordeaux) - Rational simple connectedness and Fano threefolds
26 February, 10:00-11:00
The notion of rational simple connectedness can be seen as an algebraic analogue of simple connectedness in topology. The work of de Jong, He and Starr has already produced several recent studies to understand this notion.
In a project with Laurent Gruson and Nicolas Perrin, we start the study of rational simple connectedness for Fano threefolds by explicit methods from birational geometry.
Stefano Filipazzi (UCLA) - On the connectedness principle and dual complexes for generalized pairs
23 February, 15:30-16:30
Let (X,B) be a pair (a variety with an effective Q-divisor), and let f: X -> S be a contraction with -(K_X+B) nef over S. A conjecture, known as the Shokurov-Koll\'ar connectedness principle, predicts that f^{-1}(s) intersect Nklt(X,B) has at most two connected components, where s is an arbitrary point in S and Nklt(X,B) denotes the non-klt locus of (X,B). The conjecture is known in some cases, namely when -(K_X+B) is big over S, and when it is Q-trivial over S. In this talk, we discuss a proof of the full conjecture and extend it to the case of generalized pairs. Then we apply it to the study of the dual complex of generalized log Calabi-Yau pairs. This is joint work with Roberto Svaldi.
Wahei Hara (Glasgow) - Rank two weak Fano bundle on the del Pezzo threefold of degree five
23 February, 10:00-11:00
A weak Fano bundle is a vector bundle whose projectivization has nef and big anti-canonical divisor. In this talk, we discuss a classification of rank two weak Fano bundles on the del Pezzo threefold X of degree five. In particular, we see that stable weak Fano bundles on X with trivial first Chern class are instanton in the sense of Kuznetsov. After that, using the theory of derived categories, we give resolutions of weak Fano bundles by typical vector bundles on X, and apply those resolutions to investigate the moduli spaces of weak Fano bundles. This is joint work with T. Fukuoka and D. Ishikawa.
Jesús Martínez García (Essex) - Asymptotically log del Pezzo surfaces
26 February, 11:30-12:30
Asymptotically log Fano varieties are a type of log smooth log pairs of varieties of Fano pairs introduced by Cheltsov and Rubinstein when studying the existence of Kaehler-Einstein metrics with conical singularities of maximal angle. From an MMP point of view they are strictly log canonical and as such, they do not belong to a finite number of families. However, one may hope to give a fairly explicit classification for them in low dimensions. An asymptotically log Fano variety, has an associated convex object known as the body of ample angles. Cheltsov and Rubinstein classified strongly asymptotically log del Pezzo surfaces. These are two-dimensional asymptotically log Fano varieties for which the body of ample angles is maximal around the origin. This apparently technical condition has striking consequences both for the structure and birational geometry of these surfaces, making all minimal asymptotically log del Pezzo surfaces to have rank at most two. The latter condition is what allowed Cheltsov and Rubinstein to give a full classification of asymptotically log del Pezzo surfaces. In this talk, we introduce these notions while attacking the more general problem of classifying asymptotically log del Pezzo surfaces. We further show that the body of ample angles is in fact a convex polytope.
Takuzo Okada (Saga) - Birational geometry of Fano 3-fold WCIs
24 February, 10:00-11:00
I will talk about systematic studies on birational geometry of Fano 3-fold weighted complete intersections of index 1 and codimension 2. I will especially explain recent progress on some of the remaining difficult cases, that is, a proof of birational bi-rigidity for some Fano 3-fold WCIs whose birational model admits a singular point of type cD or cE.
Erik Paemurru (Basel) - Birational geometry of sextic double solids with a compound A_n singularity
26 February, 14:00-15:00
Sextic double solids, double covers of ℙ³ branched along a sextic surface, are the lowest degree Gorenstein Fano 3-folds, hence are expected to behave very rigidly in terms of birational geometry. Smooth sextic double solids, and those which are ℚ-factorial with ordinary double points, are known to be birationally rigid. In this talk, we discuss birational geometry of sextic double solids with an isolated compound A_n singularity. I have shown that n is at most 8, and that rigidity fails for n > 3. I will illustrate this by giving some examples.
Calum Spicer (King's) - Boundedness and Fano foliations
25 February, 10:00-11:00
Fano foliations were initially introduced and studied by Araujo and Druel. I will recall some basic facts about the theory of Fano foliations, as well as explaining some pathological features of Fano foliations which obstruct the construction of the corresponding moduli space. I will then present some work in progress with Roberto Svaldi on boundedness of surface foliations and show how this provides one approach to resolving the aforementioned pathologies.