Poster session

  • Sjoerd Beentjes (Edinburgh) Donaldson-Thomas invariants of crepant resolutions of singularities The crepant resolution conjecture for Donaldson-Thomas invariants is a conjecture in enumerative geometry originating from string theory. It relates the Donaldson-Thomas generating series of a certain type of three-dimensional Calabi-Yau orbifold to that of a particular crepant resolution of its coarse moduli space. We discuss an approach to study this conjecture using derived category methods. As a partial result, we present a wall-crossing formula in (a variant of) Joyce’s motivic Hall algebra. Our formula relates the Hilbert scheme of curves on the orbifold to the Hilbert scheme of curves on the resolution. This is a first step towards potentially proving the crepant resolution conjecture for Donaldson-Thomas invariants.
  • Fabio Bernasconi (Imperial) Kawamata-Viehweg vanishing fails for log del Pezzo in char. 3 The failure of Kodaira-type vanishing theorems is the one of the main causes of pathologies in the birational geometry of varieties over fields of positive characteristic. We construct a log del Pezzo surface in characteristic 3 violating the Kawamata-Viehweg vanishing, generalizing an example due to Cascini and Tanaka in char. 2. As a consequence, we show a Kawamata log terminal 3fold singularity which is not Cohen-Macaulay in the same characteristic.
  • Matt Booth (Edinburgh) Noncommutative deformations and surface autoequivalences A threefold flop from X to Y induces a Fourier-Mukai equivalence between their derived categories. Applying the flop functor twice thus gives an autoequivalence FF of D^b(X). Recent work of Donovan and Wemyss provides an intrinsic description of FF in terms of noncommutative deformation theory, as the inverse of a certain noncommutative twist functor. Taking a cut of X by a hyperplane, one can define an analogue of the twist for surfaces, and ask whether it is an autoequivalence. In this case, one must pass to derived noncommutative deformations in order to retain enough information.
  • Pierrick Bousseau (Imperial) Tropical refined curve counting from Gromov-Witten invariants I present a new geometric interpretation of the refined counts of tropical curves introduced by Block and Göttsche. This involves generating series of log Gromov-Witten invariants and a mysterious change of variables. Reference:
  • Aurelio Carlucci (Oxford) Moduli Spaces of Stable Pairs on the Resolved Conifold  The Donaldson-Thomas strategy, developed specifically for Calabi-Yau 3-manifolds to count embedded curves, consists in looking at the Hilbert scheme of sub-curves with fixed holomorphic Euler and homology class. A drawback of the Hilbert scheme is that it admits large components with free points and is much larger than the moduli space of curves it tries to compactify. Stable pairs, introduced by Pandharipande and Thomas in 2009, provide a more manageable sheaf-theoretic approach: moreover, their moduli space can be thought of as parametrising object in the derived category of the ambient variety. The general approach in enumerative geometry is to package invariants into generating series, and prove results on inference rules to transform the series according to a change of stability parameter (wall-crossing formulæ) or the geometry of the ambient (group actions, quotients); this is done to avoid dealing with the specific geometry of the moduli space. However, there is a surprising scarcity of explicit constructions, which are much needed also in light of the interplay between PT-invariants and other invariants. The best examples are provided in the original paper by Pandharipande-Thomas and expanded by Szendrői in 2012, where also the associated mixed Hodge module is computed. The variety considered is the total space of the direct sum of two copies of the tautological bundle over the projective line, called 'resolved conifold'. Here the moduli space of stable pairs can be described explicitly as a projective scheme, leading to some interesting geometries, up to the case of sheaves supported on the double of the zero section and with holomorphic Euler characteristic 4. What I am currently thinking about is producing an equally detailed construction for the moduli space of stable pairs on the resolved conifold, with same support and Euler characteristic 5.
  • Dougal Davis (Kings) A Chevalley isomorphism for Bun_G(E) Let G be a simply connected simple algebraic group over C, with maximal torus T and Weyl group W. The classical Chevalley isomorphism is an isomorphism between the ring of G-invariant polynomial functions on G and the ring of W-invariant polynomial functions on T. We present an analogue of this theorem for the stack Bun_G(E) of principal G-bundles over an elliptic curve E, which relates line bundles on Bun_G(E) and their sections with W-invariant line bundles on the abelian variety parametrising T-bundles on E with multidegree 0.
  • Kelli Francis-Staite (Oxford) C-infinity Algebraic Geometry C-infinity rings are algebras that generalise the ring of smooth functions from a manifold to the real numbers. C-infinity rings with corners do the same for manifolds with corners. This poster motivates studying these concepts, gives an introduction to these areas, and summarises my own research on C-infinity rings and schemes with corners.
  • Roberto Fringuelli (Edinburgh) Universal moduli space of bundles on stable curves. The moduli space of semistable vector bundles over a smooth curve is a projective variety, in particular, it is compact. On the other hand, the same moduli problem over a singular curve fails to be compact. Using an idea of Gieseker, Schmitt constructed a compactification of moduli space of vector bundles on stable curves compatible on families. It seems that an analogous picture exists for any reductive group.
  • Sara Lamboglia (Warwick) Toric degenerations of flag varieties via tropical geometry
    Toric degenerations have been studied quite intensively in the last few decades and several methods have been applied. In this poster I am going to present results on the toric degenerations arising from the tropicalization of a variety. These are obtained as Gröbner degenerations associated to the initial ideals corresponding to the maximal cones of the topicalized variety. In particular I will show results for the full flag varieties of C^4 and C^5. I will then compare these degenerations with the ones arising in representation theory from string polytopes. Moreover I will present a general procedure to find toric degenerations in the cases where the initial ideal arising from a cone of the tropicalization of a variety is not prime. This is joint work with L. Bossinger, K. Mincheva and F. Mohammadi.
  • Diletta Martinelli (Edinburgh) How many are the minimal models?   Finding minimal models is the first step in the birational classification of smooth projective varieties. After it is established that a minimal model exists some natural questions arise such as: is it the minimal model unique? If not, how many are they? It is known that varieties of general type admit a finite number of minimal models. In the poster I will present a recent result (joint with Stefan Schreieder and Luca Tasin) where we prove that this number is bounded by a constant depending only on the canonical volume. I will also show that in some cases for threefolds, it is possible to compute this constant explicitly. Moreover, we prove that in any dimension minimal models of general type and bounded volume form a bounded family.
  • Enrica Mazzon (Imperial) Berkovich skeleta and essential skeleta Let X be a proper variety over a complete discretely valued field K. The geometry of its Berkovich analytification X^an may be captured by the study of some finite simplicial complexes called Berkovich skeleta. They are constructed from models of X over the ring of integers O_K. Following this idea, Mustata and Nicaise recently introduced the notion of essential skeleton of X: it is a finite simplicial complex embedded in X^an that enjoys three nice properties. Firstly, it is contained in the Berkovich skeleton of every mildly singular model. Secondly, it is a strong deformation retract of X^an by the work of Nicaise and Xu. Finally, it is a birational invariant of X. In my poster I will show how the essential skeleton reflects the geometry of the variety X. I will focus on the concrete example of the essential skeleton of the Hilbert scheme of a K3 surface S. In particular I will describe it in terms of the essential skeleton of S, only using techniques from valuation theory and the notion of weight function.
  • Claudio Onorati (Bath) Monodromy of irreducible symplectic manifolds of type O'Grady 10  We use a recent construction of Laza, Sacca and Voisin of irreducible symplectic manifolds deformation equivalent to the 10-dimensional example of O'Grady to compute a subgroup of the monodormy group of such manifolds. This subgroup is still too small, but we show where the obstructions live. Finally, we point out the direction we are walking to conclude its computation. This is a work in progress.
  • Roberto Pirisi The motivic class of the classifying stacks of G2 and the Spin groups.  
    The Grothendieck ring of varieties K0(Var/k) is the
    quotient of the free commutative group generated by the isomorphism classes  of varieties over a field k, modulo:
    - the scissor relation [X]=[U]+[V], where V is a closed subscheme of X and U is its complement.
    - the product relation [X x Y]=[X][Y]. Note that this makes[Spec(k)] a multiplicative unit.

  • This ring has been widely studied. Many important invariants, such as the Euler characteristic, factor through it.

    In the 2000s, Ekedahl defined a localization K0(Stk/k) of the Grothendieck ring, containing the classes of all algebraic stacks of finite type over k. An open question, which might be related to Noether's problem, is wether the class of the stack BG, which classifies G-torsors, satisfies the formula:
    -[BG]=1 when G is finite or [BG]=[G]^{-1} when G is infinite and connected.
    Counterexamples are known for finite G, but no counterexample is known for G an infinite connected algebraic group, and the formula is known to hold when G is special.

    We examine the question for G=G2 and G=Spin_n. We show that in the case of G=G2 and G=Spin_n, n<9 the formula holds. In general for Spin_n we show that it holds if and only if we have [BD_n]=1, where D_n is a well-known finite subgroup of Spin_n
    corresponding to the inverse image of the diagonal matrices in SO_n.
  • Andrea Ricolfi (Stavenger) A local DT/PT wall-crossing formula The DT/PT correspondence is a "wall-crossing type" formula relating Donaldson-Thomas and Pandharipande-Thomas invariants counting curves in a fixed homology class on a Calabi-Yau 3-fold. We show a "local" version of this formula, involving invariants computing the contribution of a single smooth curve. We discuss a motivic version of this formula.
  • Calum Spicer (Imperial) Foliated Mori Theory We explain some progress on developing the minimal model program for foliations on varieties. This expands on work by Brunella, McQuillan and others on the birational geometry of foliations on surfaces.
  • Alan Thompson (Cambridge) Moduli of Looijenga Pairs with Involution A Looijenga pair is a pair consisting of a rational surface along with an effective anticanonical divisor on it. I will describe moduli spaces for Looijenga pairs that admit involutions, and show that these provide a surprising generalization of Losev-Manin moduli spaces of points on rational curves.
  • Jason van Zelm (Liverpool) Pulling back cycles on the moduli space of curves to the hyperelliptic locus. The tautological ring is a particularly well understood subring of the Chow ring of the moduli space of curves. In each degree a basis of the tautological ring can be given by certain decorated boundary strata. In this poster we will explain a new technique for computing cycles on the moduli space of curves in terms of decorated boundary strata by pulling back to the hyperelliptic locus. Such descriptions are useful because they allows us to easily solve intersection theoretic questions involving these cycles.
  • Jakub Witaszek (Imperial) Frobenius liftings and toric varieties We formulate a conjecture characterising smooth projective varieties in positive characteristic whose Frobenius morphism can be lifted modulo p^2, verify it in certain special cases, and relate it to various problems in characteristic zero.