POSTECH AG SEMINAR

Seminars will be held in Math Building Room404 at 4:00-5:00 on Wednesdays. (Changes will be in RED.)
Everyone is welcome.
For more information, please contact S. Choi at
This seminar series is supported by 기하학연구센터 (GAIA, Geometry And Its Application).

2011 FALL

1. Sep 14 WED	SPEAKER: Sung Rak Choi, POSTECH
	TITLE: Base locus of divisors, LMMP, and beyond.
	ABSTRACT: I will introduce the augmented and diminished base locus of divisors. For adjoint divisors, these are precisely the loci modified by the LMMP and log canonical contraction by Abundance, respectively. Using this fact, I will generalize the result of [BDPP] which states that the dual of the pseudoeffective cone is the closure of the cone spanned by the movable curves, thereby proving a problem posed by Sam Payne.
2. Sep 21 WED	SPEAKER: Jihun Park, POSTECH
	TITLE: Birationally super-rigid Fano 3-fold hypersurfaces
	ABSTRACT: In 1979 M. Reid announced the 95 families of K3 surfaces in three dimensional weighted projective spaces . After this, A.~R.~Fletcher, who was a Ph.D. student of M.~Ried, announced the 95 families of weighted Fano threefold hypersurfaces in his Ph.D. dissertation in 1988. These are quasi-smooth hypersurfaces of degrees $d$ with only terminal singularities in weighted projective spaces\footnote{From now on, we will use weighted homogeneous coordinates $x,y,z,t$ with weights $1,a_1,a_2,a_3,a_4$ , respectively, for the weighted projective space $\mathbb{P}(1, a_1, a_2, a_3, a_4)$ .} $\mathbb{P}(1, a_1, a_2, a_3, a_4)$ , where $d=\sum a_i$ . The 95 families are determined by quadruples of non-decreasing positive integers $(a_1, a_2, a_3, a_4)$ . In late nineties, these 95 families revived and attracted birational geometers to study their properties such as birational rigidity, groups of birational automorphisms, elliptic fibration structures, and so forth. In particular, A. Corti, A. Pukhlikov, M.Reid prove that a general hypersurface in each of the 95 families of weighted Fano threefold hypersurfaces is birationally rigid. In other words, it cannot be birationally transformed into a Mori fibred spaces except itself. The goal of my talk is to explain how to prove that a quasi-smooth hypersurface in each of the 95 families is birationally rigid.
3. Sep 28 WED	SPEAKER: Donghoon Hyun, POSTECH
	TITLE: B.A.G
	ABSTRACT: Basic Algebraic Geometry
4. Oct 5 WED	SPEAKER: Youngook Choi, Yeungnam University
	TITLE: Brill-Noether divisors in the moduli space of curves
	ABSTRACT: Let $\mathcal M_g$ be the moduli space of smooth curves of genus $g$ and let $\mathcal M^r_{g,d}$ be the sublocus of $\mathcal M_g$ whose members correspond to curves admitting a linear series $g^r_d$ . If the Brill-Noether number $\rho(g,r,d)(:=g-(r+1)(g-d+r))=-1$ , then $\overline{\mathcal M}^r_{g,d}$ is an irreducible divisor in $\overline{\mathcal M}_g$ which is called a Brill-Noether divisor. These divisors play a crucial role in the birational geometry of $\mathcal M_g$ for odd genus $g\geq 23$ . In 1987, Eisenbud and Harris proved that $\mathcal M_{23}$ has Kodaira dimension $\ge 1$ by showing that $\mathcal M^1_{23,12}\neq \mathcal M^2_{23,17}$ . Farkas showed that $\mathcal M_{23}$ has Kodaira dimension $\ge 2$ by demonstrating that $\mathcal M^1_{23,12}, \mathcal M^2_{23,17}$ , and $\mathcal M^3_{23,20}$ are mutually distinct. Along this line, Ballico and Fontanari showed that $\mathcal M^r_{g,d}\neq\mathcal M^s_{g,e}$ for $r\leq 2$ and $s$ in some range. In this talk, we prove that if $g$ is odd, and $r,s,d,e$ ( $r\neq s$ ) are positive integers satisfying $\rho(g,r,d)=\rho(g,s,e)=-1$ and $e\neq 2g-2-d$ , then the supports of $\mathcal M^r_{g,d}$ and $\mathcal M^s_{g,e}$ are distinct.
5. Oct 12 WED Room404, 4-5PM	SPEAKER: Yungho Woo, POSTECH
	TITLE: Multisecant variety and homological criterion of projective variety.
	ABSTRACT: It is well known that a nondegenerate subvariety X in a projective space of degree d and codimension c > 1 has minimal degree (i.e.,d = c+1) if and only if index(X) >= c if and only if X has no multisecant c-space. In this paper we extend this result by classifying varieties with index(X) >= c - s or with no multisecant (c - s)-space for s = 1 and 2.
6. Oct 19 WED	NO SEMINAR
7. Oct 26 WED Room 210, 4:30-5:30PM	SPEAKER: Atanas Iliev, Seoul National University
	TITLE: Pencils of cubic fourfolds (common with L. Katzarkov and E. Scheidegger).
	ABSTRACT: In a recent paper arXiv:0705.1653 Maulik and Pandharipande use results of Borcherds and Kudla-Millson for O(2, 19) lattices to determine the intersection numbers of pencils of K3 of degrees 2, 4, 6 and 8 with the Noether-Lefschetz divisors in their moduli spaces. We use the analogy between moduli spaces of polarized K3 surfaces and cubic threefolds in order to obtain a similar result for pencils cubic fourfolds: finding a modular form the coefficients of which read the intersection degrees of a generic Lefschetz pencil of cubic fourfolds with the divisors of special cubic fourfolds.
8. Nov 4 Fri Room404 1:45-4:55 PM	SPEAKER: Young-Hoon Kiem, Seoul National University
	TITLE: Cotangent complex and virtual intersection theory
	ABSTRACT: This is an introductory lecture on cotangent complex, perfect obstruction theory and virtual fundamental class. Main references are Illusie’s books and Behrend-Fantechi’s papers. If time permits, I will also talk about the cosection localization principle of virtual fundamental classes.
9. Nov 9 WED Room404, 4-5PM	SPEAKER: Edward Kim, POSTECH
	TITLE: Polynomial positivity, real algebraic geometry, and symmetric optimization problems
	ABSTRACT: We discuss two problems on polynomial positivity: our questions involve a system of real polynomial inequalities over semialgebraic sets. Our polynomial inequality systems are then viewed as optimization problems which exhibit symmetries. The relaxations of these problems to tractable (and practical) classes of convex optimization problems still poses some numerical problems, so we discuss applying symmetry to attempt to solve these problems.
10. Nov 16 WED	NO SEMINAR
11. Nov 22 TUESDAY Room404, 3-4PM	SPEAKER: Sukmoon Huh, Sungkyunkwan University
	TITLE: Stability of vector bundles
	ABSTRACT: The notion of stability of was introduced for vector bundles on curves by Mumford and later generalized to sheaves on higher dimensional varieties by many people. The notion comes naturally from an algebraic point of view as well as from a gauge theoretical point view. In this talk, we will consider some situations where the stability/unstability of vector bundles on an algebraic varieties can be closely related to the geometry of the varieties.
12. Nov 30 WED Room404, 4-5PM	SPEAKER: Jinhyun Park, KAIST
	TITLE: Semi-topological cobordism for schemes
	ABSTRACT: Grothendieck (1958) shows how to define the general notion of the Chern classes for functors satisfying some axioms. It was observed by Quillen that we can obtain more diverse cohomology functors by not requiring the first Chern class map c_1 to be a homomorphism, and this led to the complex cobordism MU^* on the topological category. An analogous step was pursued by Levine and Morel for varieties to define what is called algebraic cobordism. In this talk, we report a recent work that modifies the Levine-Morel cobordism to define a theory that is an algebraic equivalence analogue on cycles. Some known results on non-finite generations of Griffiths groups, etc. are interpreted in terms of this new cobordism theory. This is a joint work with Amalendu Krishna of TIFR.
13. Dec 7 WED Room404, 4-5PM CANCELED	SPEAKER:
	TITLE:
	ABSTRACT: