# Difference between revisions of "Algebra and Algebraic Geometry Seminar Fall 2019"

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I will discuss the asymptotic non-vanishing of syzygies for products of projective spaces, generalizing the monomial methods of Ein-Erman-Lazarsfeld. This provides the first example of how the asymptotic syzygies of a smooth projective variety whose embedding line bundle grows in a semi-ample fashion behave in nuanced and previously unseen ways. | I will discuss the asymptotic non-vanishing of syzygies for products of projective spaces, generalizing the monomial methods of Ein-Erman-Lazarsfeld. This provides the first example of how the asymptotic syzygies of a smooth projective variety whose embedding line bundle grows in a semi-ample fashion behave in nuanced and previously unseen ways. | ||

+ | |||

+ | |||

+ | ===Michael Kemeny=== | ||

+ | '''The geometric syzygy conjecture''' | ||

+ | |||

+ | A famous classical result of M. Green asserts that the ideal sheaf of a canonical curve is generated by quadrics of rank four. Extending this to higher relations, one arrives at the so-called | ||

+ | Geometric Syzygy Conjecture, stating that extremal linear syzygies are spanned by those of the lowest possible rank. This conjecture further provides a geometric interpretation of Green's conjecture | ||

+ | for canonical curves. In this talk, I will outline a proof of the Geometric Syzygy Conjecture in even genus, based on combining a construction of Ein-Lazarsfeld with Voisin's approach to the study of | ||

+ | syzygies of K3 surfaces. |

## Revision as of 19:55, 9 September 2019

The seminar meets on Fridays at 2:25 pm in room B235 Van Vleck.

Here is the schedule for the previous semester, for the next semester, and for this semester.

## Algebra and Algebraic Geometry Mailing List

- Please join the AGS Mailing List to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).

## Fall 2019 Schedule

date | speaker | title | host(s) |
---|---|---|---|

September 6 | Yuki Matsubara | On the cohomology of the moduli space of parabolic connections | Dima |

September 13 | Juliette Bruce | Semi-Ample Asymptotic Syzygies | Local |

September 20 | Michael Kemeny | The geometric syzygy conjecture | Local |

September 27 | |||

October 4 | |||

October 11 | |||

October 18 | Kevin Tucker (UIC) | ||

October 25 | |||

November 1 | Michael Brown | TBD | Local |

November 8 | Patricia Klein | ||

November 15 | |||

November 22 | Daniel Corey | Topology of moduli spaces of tropical curves with low genus | Local |

November 29 | Thanksgiving Break | ||

December 6 | Reserved (Matroids Day) | ||

December 13 |

## Abstracts

### Yuki Matsubara

**On the cohomology of the moduli space of parabolic connections**

We consider the moduli space of logarithmic connections of rank 2 on the projective line minus 5 points with fixed spectral data. We compute the cohomology of such moduli space, and this computation will be used to extend the results of Geometric Langlands correspondence due to D. Arinkin to the case where the this type of connections have five simple poles on ${\mathbb P}^1$.

In this talk, I will review the Geometric Langlands Correspondence in the tamely ramified cases, and after that, I will explain how the cohomology of above moduli space will be used.

### Juliette Bruce

**Semi-Ample Asymptotic Syzygies**

I will discuss the asymptotic non-vanishing of syzygies for products of projective spaces, generalizing the monomial methods of Ein-Erman-Lazarsfeld. This provides the first example of how the asymptotic syzygies of a smooth projective variety whose embedding line bundle grows in a semi-ample fashion behave in nuanced and previously unseen ways.

### Michael Kemeny

**The geometric syzygy conjecture**

A famous classical result of M. Green asserts that the ideal sheaf of a canonical curve is generated by quadrics of rank four. Extending this to higher relations, one arrives at the so-called Geometric Syzygy Conjecture, stating that extremal linear syzygies are spanned by those of the lowest possible rank. This conjecture further provides a geometric interpretation of Green's conjecture for canonical curves. In this talk, I will outline a proof of the Geometric Syzygy Conjecture in even genus, based on combining a construction of Ein-Lazarsfeld with Voisin's approach to the study of syzygies of K3 surfaces.