During Winter 2018 the **UC Riverside Algebraic Geometry Seminar** will meet weekly, usually on Tuesdays from 12:40pm to 2:00pm in Surge 277, but the actual meeting time may vary each week. For more information you may contact Ziv Ran (ziv.ran@ucr.edu) or Jose Gonzalez (jose.gonzalez@ucr.edu). Please find our schedule below. For information about reimbursements for our visitors click here.

Date |
Speaker |
Title |
Abstract |

Thursday, January 11, 2018 Start: 1:00 PM Location: Surge 268 |
| Planning meeting. | Planning meeting. |

Tuesday, January 16, 2018 Start: 12:40 AM Location: Surge 284 |
Ziv RanUC Riverside | Learning seminar on multiplier ideals. | We will summarize some of the basic results on multiplier ideals covered last quarter and we will start a discussion on the local aspects of multiplier ideals. |

Thursday, January 25, 2018 Start: 1:00 PM Location: Surge 282 |
Zhixian ZhuUC Riverside | The subadditivity theorem. | In this talk we will introduce some applications of the restriction theorem. In particular, we will prove the subadditivity theorem for mixed multiplier ideals. |

Thursday, February 1, 2018 Start: 1:00 PM Location: Surge 284 |
Kenji HashimotoUniversity of Tokyo | Global sections of some special elliptic surfaces. | We discuss how to reconstruct an elliptic K3 surface from the data of
singular fibers. The problem is reduced to counting of global sections of
some special elliptic surfaces. |

Thursday, February 8, 2018 Start: 1:00 PM Location: Surge 282 |
Joaquin MoragaUniversity of Utah | Bounding singular surfaces via Chern numbers. | It is known that given a projective surface with mild singularities we can obtain a minimal model by contracting a sequence of curves. A natural question is which invariants of the surface can bound the number of such contractions. In this talk, I will show that a linear combination of the Chern numbers, motivated by the BMY inequality, is one of such invariants. As an application, I will discuss how to use such result to prove that certain sets of singular surfaces with bounded Chern numbers can be put together in a compact family. |

Thursday, February 15, 2018 Start: 2:00 PM Location: Surge 268 |
Zhixian ZhuUC Riverside | The subadditivity theorem. Part 2. | We will discuss some applications of the restriction theorem and we will complete the proof of the subadditivity theorem for mixed multiplier ideals. |

Thursday (*), February 22, 2018 Start: 1:00 PM Location: Surge 282 |
Neal LivesayUC Riverside | Moduli spaces of irregular singular connections. | A classical problem in mathematics is that of classifying singular differential operators. An algebro-geometric variant of this problem involves the construction of moduli spaces of connections on vector bundles over P^{1} with singularities x_{1},..., x_{k}. Locally (i.e., around a singularity x_{i}), a selection of a basis for the vector bundle induces a matrix form for the connection. The study of matrices associated to connections is analogous to the study of matrices associated to linear maps. In this talk, I will discuss a construction of moduli spaces of connections on P^{1} which are locally diagonalizable, along with recent generalizations made by C. Bremer, D. Sage, and N. Livesay. |

Thursday, March 1, 2018 Start: 2:00 PM Location: Surge 268 |
Jose GonzalezUC Riverside | Asymptotic multiplier ideals. | We will present the definition and basic properties of asymptotic multiplier ideals. These ideals measure the behavior of the linear systems of the multiples mL of a divisor L, as m goes to infinity. Useful features of this theory include some vanishing results, which we apply, as an example, to show Kollar's result on multiplicativity of plurigenera for ètale covers. |

Thursday, March 8, 2018 Start: 2:00 PM Location: Surge 268 |
Jeongseok OhKIAS and visiting scholar UC Berkeley | Localized Chern Characters for 2-periodic complexes. | The localized Chern character of a bounded complex of vector bundles is a bivariant class defined by Baum, Fulton, and MacPherson. They used such classes to prove the general Riemann-Roch theorem for singular varieties. For a two-periodic complex of vector bundles, Polishchuk and Vaintrob have constructed its localized Chern character, which is a generalization of the usual one. We discuss some properties of PV's localized Chern characters. In particular, cosection localizations defined by Kiem and Li can be expressed as these localized Chern character operations. This result is a generalization of the related work by Chang, Li, and Li. The talk is based on joint work with Bumsig Kim. |

Thursday, March 15, 2018 Start: 2:00 PM Location: Surge 268 |
Jose GonzalezUC Riverside | Examples of finitely and non-finitely generated Cox rings. | Cox rings generalize the homogeneous coordinate rings of projective spaces to varieties with finitely generated divisor class groups. When finitely generated, the underlying variety is in fact a quotient of an affine variety by a torus action and its birational geometry can be studied via geometric invariant theory. In this talk, we describe combinatorial sufficient conditions for the finite and non-finite generation of the Cox ring of the blowup at a general point of a toric surface of Picard number one. This generalizes work of Goto-Nishida-Watanabe, Kurano-Nishida and Srinivasan. We also discuss generalizations to higher dimensions and to toric surfaces with Picard number two. |

Date |
Speaker |
Title |
Abstract |

Tuesday, October 3, 2017 Start: 10:10 AM Location: Surge 268 |
| Planning meeting | Planning meeting. |

Tuesday, October 10, 2017 Start: 10:10 AM Location: Surge 268 |
Zhixian ZhuUC Riverside | Adjoint linear systems. Episode I. | Linear systems have played a central role in algebraic geometry. In this talk, we are going to study the adjoint linear system. During the first lecture, we will briefly review some basic concepts in linear systems and introduce Fujita's conjectures. There are basically three different approaches to prove the Fujita's conjectures. In the second lecture, we will introduce Sakai's cohomological method and the singularity method from a birational point of view. |

Tuesday, October 17, 2017 Start: 9:40 AM Location: Surge 268 |
Zhixian ZhuUC Riverside | Adjoint linear systems. Episode II. | Linear systems have played a central role in algebraic geometry. In this talk, we are going to study the adjoint linear system. During the first lecture, we will briefly review some basic concepts in linear systems and introduce Fujita's conjectures. There are basically three different approaches to prove the Fujita's conjectures. In the second lecture, we will introduce Sakai's cohomological method and the singularity method from a birational point of view. |

Tuesday, October 24, 2017 Start: 10:00 AM Location: Surge 268 |
Lei SongUC Riverside | Rational and Du Bois singularities | I will give an introduction to two classes of mild singularities of algebraic varieties: rational and Du Bois, and explain why they are good from the point of views of cohomology computation and deformation. I will also discuss their relation to other singularities in birational geometry. |

Tuesday, October 31, 2017 Start: 10:00 AM Location: Surge 268 |
Lei SongUC Riverside | Singularities of secant varieties | Consider the secant variety to an embedded smooth variety in some projective space. I will show if the associated line bundle is sufficiently positive, then the secant variety is Du Bois, but not rational in general. Then I will consider general normal varieties, and explain how the consideration on direct images of twisted canonical bundles from resolutions may lead to a new measurement of the singularities. |

Tuesday, November 7, 2017 Start: 10:00 AM Location: Surge 268 |
Lei SongUC Riverside | Learning seminar on multiplier ideals. Part I. | This first lecture is devoted to the construction and first properties of multiplier
ideals. We will discuss the algebraic and analytic incarnations of these ideals and present examples. |

Tuesday, November 14, 2017 Start: 10:00 AM Location: Surge 268 |
Jose GonzalezUC Riverside | Learning seminar on multiplier ideals. Part II. | We will recall the definition of multiplier ideals and discuss basic properties and examples. We will review some background material useful for grad students to follow this talk and the upcoming ones in this learning seminar. |

Tuesday, November 21, 2017 Start: 10:10 AM Location: Surge 268 |
Hwayong LeeUC Riverside | The space of lines on cyclic covers of projective space. | We study the space of lines on m-cyclic covers of projective space branched along a degree md hypersurface. As a consequence, it is a concrete example of Ran's (m_i)-contact lines to the subscheme in Fano variety and we also find the formula for counting lines with a restricted condition. This is joint work with Hosung Kim. |

Tuesday, November 21, 2017 Start: 11:10 AM Location: Surge 268 |
Ziv RanUC Riverside | Learning seminar on multiplier ideals. Part III. | We will discuss some basic properties of multiplier ideals including the independence of the log resolution used in their definition. |

Tuesday, November 28, 2017 Start: 10:00 AM Location: Surge 268 |
Ziv RanUC Riverside | Learning seminar on multiplier ideals. Part IV. | We will finish presenting the proof that the definition of multiplier ideals is independent of the choice of a log resolution, and we will start discussing the vanishing theorems for multiplier ideals. |

Tuesday, December 5, 2017 Start: 10:10 AM Location: Surge 268 |
Philip EngelHarvard University | Enumerating Triangulations | A triangulation of S^2 has non-negative curvature if every vertex has six or fewer triangles adjacent to it. Thurston showed that non-negative curvature triangulations correspond to lattice points in a moduli space of flat cone metrics on S^2. In joint work with Peter Smillie, we use an arithmetic technique of Siegel to count such lattice points. The appropriately weighted number of triangulations with 2n triangles is an explicit constant times the ninth divisor power sum of n. |

Tuesday, December 5, 2017 Start: 11:10 AM Location: Surge 268 |
Ziv RanUC Riverside | Learning seminar on multiplier ideals. Part V. | We will discuss vanishing theorems for multiplier ideals. |

Date |
Speaker |
Title |
Abstract |

Thursday, April 6, 2017 Start: 2:10 PM Location: Surge 268 |
| Planning week | No meeting this week. |

Thursday, April 13, 2017 Start: 2:10 PM Location: Surge 268 |
Daniel ChunUC Riverside | Overview of Koszul Cohomology: Applications and Current Questions | Syzygies of embedded projective varieties have historically attracted attention from many algebraists over the years. Mark Green's 1984 paper, which interprets syzygies as cohomology groups of a Koszul-type complex, made it both easier to compute them and to connect them to geometry. I will go over some applications of syzygy computations to geometric questions then introduce a current research topic about Koszul Cohomology. |

Thursday, April 20, 2017 Start: 2:10 PM Location: Surge 268 |
Omprokash DasUC Los Angeles | On the abundance problem for 3-folds in characteristic p>5 | In this talk I will explain the importance of the abundance conjecture in birational geometry, and in the classification of varieties in general. Then I will present the known cases of this conjecture in characteristic zero. Finally, I will talk about the recent advancements on this conjecture in dimension 3 and characteristic p>5. This is a joint work with Joe Waldron. |

Thursday, April 27, 2017 Start: 2:10 PM Location: Surge 268 |
Youngsu KimUC Riverside | Some commutative algebra for algebraic geometry | In this talk, we will introduce/review some commutative algebra which is often used in algebraic geometry. The primary goal of the talk is to supplement the commutative algebra part in Math 243B. If time permits, we will show that the ring C[x,y,z]/(x^2-yz) is a normal domain but not a UFD. |

Thursday, May 4, 2017 Start: 2:10 PM Location: Surge 268 |
Humberto DiazUC Riverside | Computing the class group | In this talk, I will give an overview of the basic properties of the class group of an algebraic variety, and use this to derive some tools for doing computations with the group. |

Thursday, May 11, 2017 Start: 2:10 PM Location: Surge 268 |
Youngsu KimUC Riverside | Normal rings and (discrete) valuation rings | We will continue exploring some backgrounds in algebra. This week we will revisit normal rings and then introduce (discrete) valuation rings. These rings arise often. When X is a normal k-variety, the local ring of a codimension 1 subvariety is a discrete valuation ring, DVR. |

Thursday, May 18, 2017 Start: 2:10 PM Location: Surge 268 |
Laure FlapanUC Los Angeles | Hodge Groups of Hodge Structures with Hodge Numbers (n,0,...,0,n) | One of the main tools available for proving certain cases of the Hodge conjecture for abelian varieties is to compute the Hodge groups of the weight-1 Hodge structures associated to these abelian varieties. Thus Hodge groups of abelian varieties have been extensively investigated. In this talk, we discuss generalizing these results about Hodge groups to arbitrary-weight Hodge structures with Hodge numbers (n,0,...,0,n), particularly when n is prime or twice a prime. These generalizations yield some new results about Hodge classes of 2p-dimensional abelian varieties. |

Thursday, May 25, 2017 Start: 2:10 PM Location: Surge 268 |
Fei XieUC Los Angeles | Toric surfaces over an arbitrary field | I will introduce toric varieties over arbitrary fields and classify minimal smooth projective toric surfaces. Then I will give an overview of the Merkurjev-Panin motivic category and give an explicit construction of the decomposition of toric surfaces in the motivic category into products of finite Azumaya algebras. I will explain how these Azumaya algebras determine the corresponding toric surfaces. |

Thursday, June 1, 2017 Start: 2:10 PM Location: Surge 268 |
Zhuang HeNortheastern University | New examples and non-examples of Mori Dream Spaces when blowing up toric surfaces | Mori Dream Spaces were introduced by Hu and Keel as normal, Q-factorial projective varieties whose effective cone admits a nice decomposition. As the name would indicate, Mori's minimal model program can be run for every divisor on a Mori Dream Space.
Recently there have been many studies on the question that for which integers a,b,c the blow-up of the weighted projective plane P(a,b,c) at a general point is a Mori Dream Space. In this talk, I will recall these recent work, and introduce a generalization of a result by Gonzalez and Karu in 2014. Specifically, for some toric surfaces of Picard number one, whether the blow-up is a Mori Dream Space is equivalent to countably many planar interpolation problems. I will give new examples and non-examples of Mori Dream Spaces, along with a conjecture of more non-examples, by reducing these interpolation problems. |

Thursday, June 8, 2017 Start: 2:10 PM Location: Surge 268 |
D. Blanton, T. McEldowney and A. WalkerUC Riverside | Short research presentations | Three graduate students
from our department,
working in algebraic geometry or neighboring areas,
will present the main ideas about the problems they solved or are currently working on for their theses. |

Date |
Speaker |
Title |
Abstract |

(*) Tuesday, January 10, 2017 Start: 2:10 PM Location: Surge 268 |
EveryoneUC Riverside |
Organizational Meeting | Organizational Meeting |

(*) Tuesday, January 17, 2017 Start: 2:10 PM Location: Surge 268 |
Kari VilonenNorthwestern University | Cartan theorems over a discrete valuation base | I will discuss Cartan theorems for Stein manifolds over a discrete valuation base. Although such situations are commonly considered in algebraic geometry we could not find any prior work in this direction in the context of analytic geometry. Nuclearity, a notion introduced by Groethendieck, plays an important role in our theory. This is joint work with Jari Taskinen. |

Thursday, January 26, 2017 Start: 1:10 PM Location: Surge 268 |
Carl MautnerUC Riverside | Symplectic resolutions and hypertoric varieties | This talk will survey some basic definitions and results surrounding symplectic structures in algebraic geometry. After these generalities, we will restrict our attention to a class of symplectic varieties called hypertoric varieties and discuss some of their salient features. |

Thursday, February 2, 2017 Start: 1:10 PM Location: Surge 268 |
Cristian MartinezUC Santa Barbara | Bogomolov-Gieseker Inequalities and Stability Conditions. | Given a Chern character v and an ample class H on a smooth projective complex surface, there is a distinguished open set of stability conditions so that the only semistable objects of type v are coherent sheaves that are Gieseker semistable with respect to H. Moving away from this chamber to its boundary corresponds to a contraction of the Gieseker moduli. This, for instance, accounts for all smooth MMPs on surfaces. One of the key ingredients in the construction of stability conditions on surfaces is the Bogomolov-Gieseker inequality on the Chern character of a semistable sheaf. On some threefolds a generalized inequality is satisfied by a class of "semistable" complexes, allowing for the construction of stability conditions. In this talk I will explore some of the ideas above and show a class of stable complexes violating the generalized Bogomolov-Gieseker inequality on blow-ups of smooth threefolds. |

Thursday, February 16, 2017 Start: 1:10 PM Location: Surge 268 |
Wenhao OuUCLA | Fano varieties where all pseudoeffective divisors are also numerically effective | We recall that a divisor in a smooth projective variety is said to be numerically effective (or nef) if it meets each curve with non negative intersection number, and is called pseudoeffective if it is the limit of effective Q-divisor classes. Both of these properties are ways in which a divisor can be in some sense ìpositiveî. A nef divisor is always pseudoeffective, but the converse is not true in general. A Fano varity is a special variety whose anti-canonical divisor is ample. From the Cone Theorem, it turns out that the geometry of a Fano variety is closely related to its nef divisors. In this talk, we will consider Fano varieties such that all pseudoeffective divisors are nef. Wisniewski shows that the Picard number of such a variety is at most equal to its dimension. Druel classifies these varieties when these two numbers are equal. We classify the case when the Picard number is equal to the dimension minus 1. |

Thursday, February 23, 2017 Start: 1:10 PM Location: Surge 268 |
Burt TotaroUCLA | Hodge theory of classifying spaces | The goal of this talk is to create a correspondence
between the representation theory of algebraic groups
and the topology of Lie groups. The idea is to study
the Hodge theory of the classifying space of a reductive
group over a field of characteristic p, the case
of characteristic 0 being well known. The approach
yields new calculations in representation theory,
motivated by topology. |

Thursday, March 2, 2017 Start: 1:10 PM Location: Surge 268 |
Stefano VidussiUC Riverside | Kodaira fibrations and surface group extensions | In this talk I will discuss some constraints on surface group-by-surface group extensions to be fundamental groups of Kodaira fibrations. |

Thursday, March 9, 2017 Start: 1:10 PM Location: Surge 268 |
Christopher LyonsCalifornia State University, Fullerton | Applications of some double covers of a class of surfaces of general type | We will focus on complex algebraic surfaces with invariants p_g=q=1 and K^2=2, an interesting class of surfaces of general type first classified in the late 1970s by Bombieri-Catanese and Horikawa. Inspired by work of Ishida, we describe how to obtain polynomial equations for unramified double covers of these surfaces.
These more accessible double covers allow one to obtain results about the original surfaces themselves. First we will discuss how one may obtain (via zeta functions) an explicit surface with p_g=q=1, K^2=2 having minimal Picard number. This result and others then contribute towards proofs about the larger family of surfaces with these invariants, such as a big monodromy theorem and the Tate Conjecture in characteristic zero.
This is joint work with Paul Lewis. |

Thursday, March 16, 2017 Start: 1:10 PM Location: Surge 268 |
Humberto DiazUC Riverside | Algebraic Cycles mod n. Episode II. | In this talk, I will give an overview of the group of algebraic cycles on a smooth projective variety and some infinitude results for this group. I will also report on work in progress for a new case of infinitude with mod n coefficients. |

Date |
Speaker |
Title |
Abstract |

Tuesday, October 04, 2016 Start: 2:10 PM Location: Surge 282 (*) |
Ziv RanUC Riverside |
Filling Groovy: Generic Projections | We consider a general fibre of given length in a generic projection of a variety. Under the assumption that the fibre is of local embedding dimension 2 or less, an assumption which can be checked in many cases, we prove that the fibre is reduced and its image on the projected variety is an ordinary multiple point. |

Tuesday, October 11, 2016 Start: 2:10 PM Location: Surge 268 |
Daniel ChunUC Riverside | Non-vanishing of syzygies and secant constructions, Part I | After an introduction into the world of syzygies, we consider various techniques to explicity compute syzygies of projective varieties of large degree. Although we will look at some vanishing results, our focus will primarily be on proving non-vanishing of syzygies through an induction involving secant constructions, which is an approach developed by Lazarsfeld and Ein in their paper "Asymptotic syzygies of algebraic varieties." |

Tuesday, October 18, 2016 Start: 2:10 PM Location: Surge 268 |
Daniel ChunUC Riverside | Non-vanishing of syzygies and secant constructions, Part II | After an introduction into the world of syzygies, we consider various techniques to explicity compute syzygies of projective varieties of large degree. Although we will look at some vanishing results, our focus will primarily be on proving non-vanishing of syzygies through an induction involving secant constructions, which is an approach developed by Lazarsfeld and Ein in their paper "Asymptotic syzygies of algebraic varieties." |

Tuesday, October 25, 2016 Start: 2:10 PM Location: Surge 268 |
Daniel ChunUC Riverside | Non-vanishing of syzygies and secant constructions, Part III | After an introduction into the world of syzygies, we consider various techniques to explicity compute syzygies of projective varieties of large degree. Although we will look at some vanishing results, our focus will primarily be on proving non-vanishing of syzygies through an induction involving secant constructions, which is an approach developed by Lazarsfeld and Ein in their paper "Asymptotic syzygies of algebraic varieties." |

Tuesday, November 01, 2016 Start: 2:10 PM Location: Surge 268 |
Hwayoung LeeVisiting UC Riverside |
Mirror symmetry on complete intersection K3 surfaces | This talk is based on joint work with Kenji Hashimoto and Kazushi Ueda.
On complete intersection K3 surfaces in Weighted projective 4-fold, we show that
the duality by (an analogue of) transposition mirror construction of Berglund-Huebsch is equal to Dolgachev's mirror symmetry. |

Tuesday, November 08, 2016 Start: 2:10 PM Location: Surge 268 |
Humberto DiazUC Riverside | Algebraic Cycles mod n | In this talk, I will give an overview of the group of algebraic cycles on a smooth projective variety and some infinitude results for this group. I will also report on work in progress for a new case of infinitude with mod n coefficients. |

Tuesday, November 15, 2016 Start: 2:10 PM Location: Surge 268 |
Youngsu KimUC Riverside | Castelnuovo-Mumford Regularity; a counter example of McCullough and Peeva | Castelnuovo-Mumford regularity is an invariant of a graded-module over a polynomial ring, which controls the maximum shifts of the graded syzygies. There has been a long standing conjecture by Eisenbud-Goto. Roughly speaking, it says that the regularity of a graded ideal is bounded by its multiplicity (or degree). There have been several positive results in both commutative algebra and algebraic geometry. Recently, Jason McCullough and Irena Peeva announced a counter-example to the regularity conjecture. In this talk, we briefly review the history of the conjecture and the construction of their example. |

Tuesday, November 22, 2016 Start: 2:10 PM Location: Surge 268 |
Stefano VidussiUC Riverside |
The slope of surfaces of Albanese dimension one | The geography of minimal surfaces of general type is constrained by the Bogomolov-Miyaoka-Yau line and the Noether line. Sommese [84] proved that the set of attainable slopes is dense in the region determined by these lines. In this talk we will discuss how such result can be recast using only surfaces of Albanese dimension one. |

Tuesday, November 29, 2016 Start: 2:10 PM Location: Surge 268 |
Jose GonzalezUC Riverside |
Some non-finitely generated Cox rings | Cox rings generalize the homogenous coordinate rings of toric varieties. The main problem in the theory of Cox rings is to determine whether they are finitely generated. We overview some non-finite generation results for the Cox rings of weighted projective planes blown up at a point, which extend work of Goto, Nishida and Watanabe. These results have been used by several authors to study the Cox rings of the moduli space of pointed genus zero curves. |