# 2017-18

## Contents |

# Algebra and Algebraic Geometry seminar at UBC

Except when noted otherwise, the seminar meets **Monday 4-5pm** in **MATH 126**.

## Announcements

Talks and schedule changes are announced on the alggeom mailing list. Please email the organizers to be added to the list.

## Organizers

Ádám Gyenge (agyenge@math.ubc.ca), Fei Hu (fhu@math.ubc.ca)

## Schedule

### 2017/18 Winter Term 2

Date | Speaker | Title/Abstract |
---|---|---|

2018-01-08 | Joontae Kim (Seoul National University) | Title: Wrapped Floer homology of real Lagrangians and volume growth of symplectomorphisms |

Abstract: Floer homology has been a central tool to study global aspects of symplectic topology, which is based on pseudoholomorphic curve techniques proposed by Gromov. In this talk, we introduce a so-called wrapped Floer homology. Roughly speaking, this is a certain homology generated by intersection points of two Lagrangians and its differential is given by counting solutions to perturbed Cauchy-Riemann equation. We investigate an entropy-type invariant, called the slow volume growth, of certain symplectomorphisms and give a uniform lower bound of the growth using wrapped Floer homology. We apply our results to examples from real symplectic manifolds, including A_k-singularities and complements of a complex hypersurface. This is joint work with Myeonggi Kwon and Junyoung Lee. | ||

2018-01-15 | CANCELED - Nicolas Addington (Oregon) | Title: CANCELED - Exoflops |

2018-01-22 | Steven Rayan (Saskatchewan) | Title: Asymptotic geometry of hyperpolygons |

Abstract: Nakajima quiver varieties lie at the interface of geometry and representation theory. I will discuss a particular example, hyperpolygon space, which arises from star-shaped quivers. The simplest of these varieties is a noncompact complex surface admitting the structure of an "instanton", and therefore fits nicely into the Kronheimer-Nakajima classification of ALE hyperkaehler 4-manifolds, which is a geometric realization of the McKay correspondence for finite subgroups of SU(2). For more general hyperpolygon spaces, we speculate on how this classification might be extended by studying the asymptotic geometry of the variety. In moduli-theoretic terms, this involves driving the stability parameter for the quotient to an irregular value. This is joint work in progress with Harmut Weiss, building on previous work with Jonathan Fisher. | ||

2018-01-29 | Federico Scavia (UBC) | Title: Essential dimension of representations of algebras |

Abstract: The essential dimension of an algebraic object is the minimal number of independent parameters one needs to define it. I will explain how the representation type of a finitely-generated algebra (finite, tame, wild) is determined by the essential dimension of the functors of its n-dimensional representations and I will introduce new numerical invariants for algebras. I will then illustrate the theorem and explicitely determine the invariants in the case of quiver algebras. | ||

2018-02-05 | Roberto Pirisi (UBC) | Title: A birational Gabriel's theorem |

Abstract: A famous theorem by Gabriel asserts that two Noetherian schemes X, Y are isomorphic if and only if the categories Coh(X), Coh(Y) are isomorphic. This theorem has been extended in many directions, including algebraic spaces and stacks (if we consider the monoid structure given by tensor product). One more idea to extend the theorem is the following: let X be a scheme of finite type over a field k, and consider the subcategory of Coh(X) given by sheaves supported in dimension at most d-1. We can form the quotient of Coh(X) by this subcategory, which we will call C_d(X). This category should contain enough information to describe the geometry of X "up to subsets of dimension d-1". In a joint work in progress with John Calabrese, we show that this is indeed true, i.e. to any isomorphism f: C_d(Y) ---> C_d(X) we can associate an isomorphism f': U---> V, where U and V are open subset respectively of X and Y whose complement have dimension at most d-1. Additionally, this isomorphism is unique up to subsets of dimension at most d-1. As a corollary of this result, we show that the automorphisms of C_d(X) are in bijection with the set {"automorphisms of X up to subsets of dimension d-1"} x {"line bundles on X up to subsets dimension d-1"}. (joint w/ J. Calabrese). | ||

2018-02-12 | No seminar (Family day) | |

2018-02-19 | No seminar (Midterm break) | |

2018-02-26 | Nathan Ilten (SFU) | Title: Fujita's Freeness Conjecture for Complexity-One T-Varieties |

Location: MATX1118 | Abstract: Fujita famously conjectured that for a $d$-dimensional smooth projective variety $X$ with ample divisor $H$, $mH+K_X$ is base point free whenever $m\geq d+1$. I will discuss recent joint work with Klaus Altmann in which we show this conjecture is true whenever $X$ admits an effective action by a torus of dimension $d-1$. | |

2018-03-05 | Jarod Alper (Washington) | Title: Quotients of algebraic varieties |

Abstract: In this talk, we will address the following question: given an algebraic group $G$ acting on a variety $X$, when does the quotient $X/G$ exist? We will provide an answer to this question in the case that $G$ is reductive by giving necessary and sufficient conditions for the quotient to exist. We will discuss various applications to equivariant geometry, moduli problems and Bridgeland stability. | ||

2018-03-12 | Nikita Karpenko (UAlberta) | Title: On generic flag varieties of Spin(11) and Spin(12) |

Abstract: Let X be the variety of Borel subgroups of a split semisimple
algebraic group G over a field, twisted by a generic G-torsor. Conjecturally, the canonical epimorphism of the Chow ring CH(X) onto the associated graded ring GK(X) of the topological filtration on the Grothendieck ring K(X) is an isomorphism. We prove new cases G=Spin(11) and G=Spin(12) of this conjecture. On an equivalent note, we compute the Chow ring CH(Y) of the highest orthogonal grassmannian Y for the generic 11- and 12-dimensional quadratic forms of trivial discriminant and Clifford invariant. In particular, we describe the torsion subgroup of the Chow group CH(Y) and determine its order which is equal to 16 777 216. On the other hand, we show that the Chow group of 0-cycles on Y is torsion-free. | ||

2018-03-19 | Nicholas Proudfoot (Oregon) | Title: The quantum Hikita conjecture |

Abstract: The Hikita conjecture relates the cohomology ring of a symplectic resolution to the coordinate ring of another such resolution. I will explain this conjecture, and present a new version of the conjecture involving the quantum cohomology ring. There will be an emphasis on explicit examples. | ||

2018-03-26 | Carl Mautner (UC Riverside) | Title: Springer theory and hypertoric varieties |

Abstract: The nilpotent cone has very special geometry which encodes interesting representation theoretic information. It is expected that many of its special properties have analogues for general “symplectic singularities.” This talk will discuss one such analogy for a class of symplectic singularities called hypertoric varieties. The main result, joint with T. Braden, can be described as a duality between nearby and vanishing cycle sheaves on Gale dual hypertoric varieties. | ||

2018-04-02 | No seminar (Easter Monday) | |

Extra: 2018-04-05 | Frances Kirwan (Oxford) | Title: Hyperkahler implosion |

Location: MATH 126 | Abstract: The hyperkahler quotient construction (introduced by Hitchin et al in the 1980s) allows us to construct new hyperkahler spaces from suitable group actions on hyperkahler manifolds. This construction is an analogue of symplectic reduction (introduced by Marsden and Weinstein in the 1970s), and both are closely related to the quotient construction for complex reductive group actions in algebraic geometry provided by Mumford's geometric invariant theory (GIT). Hyperkahler implosion is in turn an analogue of symplectic implosion (introduced in a 2002 paper of Guillemin, Jeffrey and Sjamaar) which is related to a generalised version of GIT providing quotients for non-reductive group actions in algebraic geometry. |

### 2017/18 Winter Term 1

Date | Speaker | Title/Abstract |
---|---|---|

2017-09-11 | Fei Hu (UBC) | Title: Dynamics on automorphism groups of compact Kähler manifolds |

Abstract: Given a compact Kähler manifold X and a biholomorphic self-map g of X, the topological entropy of g plays an important role in the study of dynamical system (X, g). In this talk, I first talk about a generalization of a surface result, that is, a parabolic automorphism of a compact Kähler surface preserves an elliptic fibration, to hyperkähler manifolds. We give a criterion for the existence of equivariant fibrations on ‘certain’ hyperkähler manifolds from a dynamical viewpoint. Next, I will generalize a finiteness result for the null-entropy subset of a commutative automorphism group due to Dinh–Sibony (2004), to arbitrary virtually solvable groups G of maximal dynamical rank. This is based on joint work with T.-C. Dinh, J. Keum, and D.-Q. Zhang. | ||

2017-09-18 | Eugene Gorsky (UC Davis) | Title: Knot invariants and Hilbert schemes |

Abstract: I will discuss some recent results and conjectures relating knot invariants (such as HOMFLY-PT polynomial and Khovanov-Rozansky homology) to algebraic geometry of Hilbert schemes of points on the plane. All notions will be introduced in the talk, no preliminary knowledge is assumed. This is a joint work with Andrei Negut and Jacob Rasmussen. | ||

2017-09-25 | Ádám Gyenge (UBC) | Title: A power structure over the Grothendieck ring of geometric dg categories |

Abstract: The notion of a power structure is closely related to that of a lambda ring. It is a powerful way to encode operations on certain generating functions. Gusein-Zade, Luengo, and Melle-Hernandez have defined a power structure over the Grothendieck ring of varieties. I will discuss an analog of this on a version of the Grothendieck ring of pretriangulated categories, whose elements represent enhancements of derived categories of coherent sheaves on varieties. | ||

2017-10-02 | Samuel Bach (UBC) | Title: Derived algebraic geometry and L-theory |

Abstract: L-theory is often dubbed as "the K-theory" of quadratic forms. It has been used in a crucial way in surgery theory, to determine if two manifolds are cobordant. I will explain how it is easily defined in the derived setting by considering "derived" quadratic forms, and how I have used derived algebraic geometry to prove a rigidity result for L-theory. This will give an application of derived methods to a non-derived problem. | ||

2017-10-09 | No seminar (Thanksgiving) | |

2017-10-16 | Klaus Altmann (FU Berlin) | Title: Infinitesimal qG-deformations of cyclic quotient singularities |

Abstract: The subject of the talk is two-dimensional cyclic quotients, i.e. two-dimensional toric singularities. We introduce the classical work of Koll'ar/Shephard-Barron relating the components of their deformations and the so-called P-resolutions, we give several combinatorial descriptions of both gadgets, and we will focus on two special components among them - the Artin component allowing a simultaneous resolution and the qG-deformations preserving the Q-Gorenstein property. That is, it becomes important that several (or all) reflexive powers of the dualizing sheaf fit into the deformation as well. We will study this property in dependence on the exponent r. While the answers are already known for deformations over reduced base spaces (char = 0), we will now focus on the infinitesimal theory. (joint work with János Kollár) | ||

2017-10-23 | Angelo Vistoli (SNS Pisa) | Title: Chow rings of some stacks of smooth curves |

Abstract: There is by now an extensive theory of rational Chow rings of stacks of smooth curves. The integral version of these Chow rings is not as well understood. I will survey what is known, including some recent developments. | ||

2017-10-30 | Jim Bryan (UBC) | Title: Donaldson-Thomas invariants of the banana manifold and elliptic genera. |

Abstract: The Banana manifold (or bananafold for short), is a compact Calabi-Yau threefold X which fibers over P^1 with Abelian surface fibers. It has 12 singular fibers which are non-normal toric surfaces whose torus invariant curves are a banana configuration: three P^1’s joined at two points, each of which locally look like the coordinate axes in C^3. We show that the Donaldson-Thomas partition function of X (for curve classes in the fibers) has an explicit product formula which, after a change of variables is the same as the generating function for the equivariant elliptic genera of Hilb(C^2), the Hilbert scheme of points in the plane. | ||

2017-11-06 | Jesse Wolfson (UC Irvine) | Title: The Theory of Resolvent Degree - After Hamilton, Sylvester, Hilbert, Segre and Brauer |

Abstract: Resolvent degree is an invariant of a branched cover which quantifies how "hard" is it to specify a point in the cover given a point under it in the base. Historically, this was applied to the branched cover P^n/S_{n-1} -> P^n/S_n, from the moduli of degree n polynomials with a specified root to the moduli of degree n polynomials. Classical enumerative problems and congruence subgroups provide two additional sources of branched covers to which this invariant applies. In ongoing joint work with Benson Farb, we develop the theory of resolvent degree as an extension of Buhler and Reichstein's theory of essential dimension. We apply this theory to systematize an array of classical results and to relate the complexity of seemingly different problems such as finding roots of polynomials, lines on cubic surfaces, and level structures on intermediate Jacobians. | ||

2017-11-13 | No seminar (Remembrance day) | |

2017-11-20 | Hirotachi Abo (Idaho) | Title: Equations for surfaces in projective four-space |

Abstract: This talk is concerned with the question of the minimal number of equations necessary to define a given projective variety scheme-theoretically. Every hypersurface is cut out by a single polynomial scheme-theoretically (also set-theoretically and ideal theoretically). Therefore the question is more interesting if a variety has a higher codimension. In this talk, we focus on the case when the codimension is two. If a variety in projective n-space has codimension two, then the minimal number of polynomials necessary to cut out the variety scheme-theoretically is between 2 and n+1. However the varieties cut out by fewer than n+1, but more than 2 polynomials seem very rare. The main goal of this talk is to discuss conditions for a non-singular surface in projective four-space to be cut out by three polynomials. | ||

Extra: 2017-11-21 | Shamil Asgarli (Brown) | Title: The Picard group of the moduli of smooth complete intersections of two quadrics |

Abstract: We study the moduli space of smooth complete intersections of two quadrics by relating it to the geometry of the singular members of the corresponding pencil. We give a new description for this parameter space by using the fact that two quadrics can be simultaneously diagonalized. Using this description we can compute the Picard group, which always happens to be cyclic. For example, we show that the Picard group of the moduli stack of smooth degree 4 Del Pezzo surfaces is Z/4Z.
This is a joint work with Giovanni Inchiostro. | ||

Extra: 2017-11-23 | Frank Sottile (Texas A&M) | Title: Irrational Toric Varieties |

Location: MATX 1102 | Abstract: Classical toric varieties come in two flavours: Normal toric varieties are given by rational fans in R^n. A (not necessarily normal) affine toric variety is given by finite subset A of Z^n. When A is homogeneous, it is projective. Applications of mathematics have long studied the positive real part of a toric variety as the main object, where the points A may be arbitrary points in R^n. For example, in 1963 Birch showed that such an irrational toric variety is homeomorphic to the convex hull of the set A.
Recent work showing that all Hausdorff limits of translates of irrational toric varieties are toric degenerations suggested the need for a theory of irrational toric varieties associated to arbitrary fans in R^n. These are R^n_>-equivariant cell complexes dual to the fan. Among the pleasing parallels with the classical theory is that the space of Hausdorff limits of the irrational projective toric variety of a finite set A in R^n is homeomorphic to the secondary polytope of A. | |

2017-11-27 | Daniel Litt (Columbia) | Title: Arithmetic representations of fundamental groups |

Abstract: Let X be an algebraic variety over a field k. Which representations of pi_1(X) arise from geometry, e.g. as monodromy representations on the cohomology of a family of varieties over X? We study this question by analyzing the action of the Galois group of k on the fundamental group of X, and prove several fundamental structural results about this action.
As a sample application of our techniques, we show that if X is a normal variety over a field of characteristic zero, and p is a prime, then there exists an integer N=N(X,p) such that any non-trivial p-adic representation of the fundamental group of X, which arises from geometry, is non-trivial mod p^N. |