Schedule-at-a-Glance

6 PRIMA 2013 Abstractsruled or finite over their centres. We have a proof of theconjecture that depends adds a few other conditions. Weshow that such domains over finite fields are finite overtheir centres. Then a deformation theory argument fororders over surfaces shows that we have Artin’s conjecturedclassification. This is work in progress with JasonBell.Geometry and syzygies in the first linearstrandSijong KwakKorea Advanced Institute of Science and Technology, Koreasjkwak@kaist.ac.krProperty N d,p , d ≥ 2 for algebraic sets is defined (dueto Eisenbud-Green-Hulek-Popescu) as follows: the j-thsyzygies of the homogeneous coordinate ring are generatedby elements of degree ≤ d−1+j for 1 ≤ j ≤ p. Whend = 2, linear syzygies of quadratic schemes have been focusedfor a long time. In this talk, we consider upperbounds and lower bounds of higher linear syzygies in thefirst linear strand of Betti tables for projective varietiesin arbitrary characteristic. For this purpose, we establishfundamental inequalities which govern the relationsbetween the graded Betti numbers of an algebraic set Xand those of its inner projection X q in the first linearstrand. We obtain some natural sharp upper bounds andlower bounds for linear syzygies of any non-degenerateprojective variety using these inequalities. We also classifywhat the extremal case and next-to-extremal caseare. From the viewpoint of ’syzygies’, this is a generalizationof Castelnuovo and Fano’s results on the numberof quadrics containing a given variety.Categorification of Donaldson-Thomas invariantsand Gopakumar-Vafa invariantsJun LiStanford University, USAjli@math.stanford.eduFor a projective Calabi-Yau threefold, we can form themoduli of stable sheaves with prescribed Chern classes.The Donaldson-Thomas invariants of this Calabi-Yauthreefolds are virtual degrees of these moduli spaces.In this talk, we will show that each of such modulispaces admits perverse sheaves whose Euler classes arethe Donaldson-Thomas invariants associated to the modulispaces. Using such sheaves, we propose a new definitionof Gopakumar-Vafa invariants of Calabi-Yau threefolds.This is a joint work with Young-Hoon Kiem.CompactsAmnon NeemanThe Australian National University, AustraliaAmnon.Neeman@anu.edu.auFor a flat map f : X −→ Y of noetherian schemes onemay define functors Lf ∗ , Rf ∗, f × and f ! on appropriatederived categories of quasicoherent sheaves. In two recentpapers, one by Avramov and Iyengar and the secondby Avramov, Iyengar, Lipman and Nayak, the authorsproved some remarkable formulas relating f ! , f × and certainHochschild homology objects. I will discuss a newapproach to the results in terms of a map ψ : f × −→ f ! .Smooth quartic K3 surfaces and CremonatransformationsKeiji OguisoOsaka University, Japan and Korea Institute for AdvancedStudy, Koreaoguiso@math.sci.osaka-u.ac.jpWe discuss about the following question of Gizatullin:Question. Is an automorphism g of a smooth quarticK3 surface S ⊂ P 3 derived from some birational automorphism,i.e., some Cremona transformation of the ambientspace P 3 ? More precisely, first I explain the followingnegative result: Theorem 1. There is a smooth K3 surfaceS of Picard number 2 such that (i) Aut (S) ≃ Z and(ii) No element Aut (S) other than id S is derived fromBir (P 3 ) in any embedding of S into P 3 . Then, I explainthe following positive result: Theorem 2. Thereis a smooth K3 surface S of Picard number 2 such that(i) Aut (S) ≃ Z and (ii) Aut (S) is derived from Bir (P 3 )in any embedding of S into P 3 . As I will explain inmy talk, examples in Theorem 2 are the ones found in acompletely different context. However, they turn out tobe very closely related with old works of Cayley, Snyderand Sharpe, as it is discovered by Festi, Garbagnati, vanGeemen and vam Luijk quite recently. Being based onall of their works, I describe the automorphism group ofthese S both explicitly algebraically in terms of homogeneouscoordinates and explicitly geometrically in terms ofSarkisov link.The Hilbert scheme of points and extensionsof local fieldsTakehiko YasudaOsaka University, Japantakehikoyasuda@math.sci.osaka-u.ac.jpTwo formulas very similar to each other appear in totallydifferent contexts, the Hilbert scheme of points and Bhargava’sformula for extensions of local fields. In this talk,I will explain the similarity as a special case of the wildMcKay correspondence. The talk is based on a joint workwith Melanie Matchett Wood.Positivity of log canonical divisors andMori/Brody hyperbolicityDe-Qi ZhangNational University of Singapore, Singaporematzdq@nus.edu.sgLet X be a complex projective variety of dimension n andD a reduced divisor with a decomposition D = ∑ ri=1 D i,where the D i ’s are reduced Cartier but not necessarilyirreducible. The pair (X, D) is called Brody hyperbolic,respectively Mori hyperbolic, with respect to the decompositionif neither X\D nor (∩ i∈I D i )\(∪ j∈J D j ) containsa non-constant holomorphic image, respectively algebraicimage, of C for every partition of {1, . . . , r} = I ∐ J.Assuming that the singularities of the pair (X, D) aresufficiently mild, we show that the log canonical divisorK X +D is numerically effective in the case of Mori hyperbolicityand that K X + D is ample provided that eithern < 4 and D is non-empty or at least n−2 of the D i ’s areample in the case of Brody hyperbolicity. Our proof alsogives simple geometric criteria for the log canonical divisorK X + D to be numerically effective or to be ample inthe above (respective) cases and, under weaker geometricconditions, for K X + D to be pseudo-effective. This is ajoint work with Steven Lu.Special Session 3Algebraic Topology and Related TopicsHomotopy colimits and commutative elementsin Lie groupsAlejandro AdemUniversity of British Columbia, Canadaadem@pims.math.ca