Geometry of Cubics

Introduction: Cubics have been intensively studied from various aspects. Among all questions, determining its rationality is one of the most interesting open problems in algebraic geometry. The purpose of this semester’s seminar is to understand the related work, from Clemens-Griffiths’s proof of the non-rationality of cubic threefolds using Hodge theory and its extension to higher dimension (by Beauville, Hassett, O’Grady, Voisin) to the more recent approach of derived category introduced by Kzunetsov.  

The activity will be weekly held alternatively in THU and PKU. The first meeting (on 2015.3.11) will have an organizational part which assigns the materials. Any people who are interested can contact one of the organizers. 

Organizers: Baohua Fu (CAS), Eduard Looijenga (THU), Chenyang Xu (PKU)

Schedule: Wednesday (3:20pm-5pm)

Classrooms: BICMR 82J04, THU Room 3 (2nd floor)

Contents and Speakers:

1. Cubic threefolds.

March 11 (PKU) Geometry of Fano scheme ［2］Speaker: Evegeny Mayankiy 

March 18 (THU) Intermediate Jacobians: Principal Polarized Abelian Varieties and its Theta divisor ［4］ Speaker: Jinsong Xu

April 1  (PKU) The proof of nonrationality by Clemens-Griffiths ［4］ Speaker: Chenyang Xu

April 8  (Thu) The proof of nonrationality by Clemens-Griffiths ［4］ Speaker: Chenyang Xu

2. Cubic Fourfolds.

April 15 (PKU) Geometry of Fano scheme: Hyperkahler manifold. [3] Speaker: Zhiwei Zheng

April 22 (THU) Hodge theoretic study and its relationship with K3. [6]  Speaker: Ze Xu

April 29 (PKU) Hodge theoretic study and its relationship with K3. [6]  Speaker: Ze Xu

May 6 (THU) Some other constructions of Hyperkahler. [5,8,9]  Speaker: Baohua Fu

May 13 (PKU) Rationality question via Hodge theory: Hassett's conjecture, [6] Speaker: Wenfei Liu

May 27  (THU) Derived Category: Kuznetsov's approach [7] Speaker: Peng Sun

June 3  (PKU) A comparison results [1]. Speaker: Jiaming Chen

Materials:

1. Addington, Nicolas; Thomas, Richard Hodge theory and derived categories of cubic fourfolds. Duke Math. J. 163 (2014), no. 10, 1885–1927. 

2. Altman, Allen B.; Kleiman, Steven L. Foundations of the theory of Fano schemes. Compositio Math. 34 (1977), no. 1, 3–47.

3. Beauville, Arnaud; Donagi, Ron La variété des droites d'une hypersurface cubique de dimension. (French) [The line manifold of a cubic fourfold] C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 14, 703–706. 

4. Clemens, C. Herbert; Griffiths, Phillip A. The intermediate Jacobian of the cubic threefold. Ann. of Math. (2) 95 (1972), 281–356.

5. Debarre, Olivier; Voisin, Claire Hyper-Kähler fourfolds and Grassmann geometry. J. Reine Angew. Math.649 (2010), 63–87. 

6. Hassett, Brendan Special cubic fourfolds. Compositio Math. 120 (2000), no. 1, 1–23.

7. Kzunetsov, Alexander derived categories of cubic fourfolds. Cohomological and geometric approaches to rationality problems, 219–243, Progr. Math., 282, Birkhäuser Boston, Inc., Boston, MA, 2010. 

8. O'Grady, Kieran G. A new six-dimensional irreducible symplectic variety. J. Algebraic Geom. 12 (2003),no. 3, 435–505. 

9. O'Grady, Kieran G. Desingularized moduli spaces of sheaves on a K3 J. Reine Angew. Math. 512(1999), 49–117.