SMUGA, Working seminar in Algebraic Geometry

Academimc year 2012/2013

S.M.U.G.A. = Seminarium Młodych Umysłów Geometrii Algebraicznej (not quite, something like: "seminar of young algebro-geometric minds").
smuga = Polish word for "streak".

Organisers: Weronika Buczyńska and Jarosław Buczyński

SMUGA is a working seminar in Algebraic Geometry running in Warsaw (either at MIMUW or at IMPAN). It is intended for students of Master program, PhD students, postdocs and professors interested in Algebraic Geometry. Everyone is welcome to attend and suggest a topic (also for future).

The idea of the seminar is to understand in detail a very specific problem, subject, article, or series of related articles.

We meet on Thursdays 10:15--11:45 at MIMUW, just before the Algebraic Geometry Seminar. The classroom is 5420.

Schedule:

When?Who?What?
04.10.2012Joachim Jelisiejew Thm 1.1 in Białynicki-Schinzel
11.10.2012Sofia Tirabassi Determinantal equations 1: Intro, Sect. 2, Koszul cohomology)
18.10.2012Sofia Tirabassi Determinantal equations 2: Sect. 2, Koszul cohomology continued, Castelnuovo Mumford regularity and multigraded regularity
25.10.2012Weronika Buczyńska Determinantal equations 3: Sect. 3, Proof of the main theorem
01.11.2012****no meeting ****
08.11.2012Oskar Kędzierski Cox rings 1: Cox rings of K3 surfaces
15.11.2012Jarek Buczyński Determinantal equations 4: Regularity, bounds on Hilbert function, and persistence with applications to cactus varieties; why we are interested in the "Determinantal Equations..."
22.11.2012Marysia Donten-BuryCox rings 2: the total coordinate ring of a toric variety; construction of toric varieties as quotients of torus actions. Based on Chapters 5.1 and 5.2 of the book "Toric varieties" by Cox, Little, Schenck.
29.11.2012Jarek Wiśniewski Cox rings 3: Cox rings from historical perspectives; its relation with group actions and quotients; examples.
06.12.2012Elisa PostinghelCox rings 4: Mori Dream Spaces
13.12.2012Jarek Wiśniewski Cox rings 5: MMP and Cox rings
20.12.2012****no meeting ****
10.01.2013Oskar Kędzierski Cox rings 6, begining of the ADHL book
17.01.2013Oskar Kędzierski Cox rings 7, following ADHL. Canonical toric embedding of a variety arising from the bunched ring construction; Geometry of such variety.
24.01.2013Marysia Donten-BuryCox rings 8, quotient characterization of the Cox ring (following ADHL, section I.6). Probably also an example: bunched rings of resolutions of surface quotient singularities.
31.01.2013****no meeting****Exam session
7.02.2013****no meeting****Exam session
14.02.2013****no meeting****Winter holidays
21.02.2013Jarek BuczyńskiMaps of toric varieties in Cox coordinates 1
28.02.2013Jarek BuczyńskiMaps of toric varieties in Cox coordinates 2: proof of existence of description and an algorithm to find it
7.03.2013Weronika BuczyńskaGeramita's Lecture Notes 1
14.03.2013Elisa Postinghel Geramita's Lecture Notes 2
21.03.2013Agnieszka BodzentaGeramita's Lecture Notes 3
28.03.2013****no meeting****Easter holidays
04.04.2013Joachim JelisiejewGeramita's Lecture Notes 4
11.04.2013Stefano Urbinati Geramita's Lecture Notes 5
18.04.2013Ada Boralevi Geramita's Lecture Notes 6
25.04.2013Zbigniew Jelonek Affine manifolds with exotic structures
2.05.2013****no meeting****free week
9.05.2013Grzegorz Kapustka Geramita's Lecture Notes 7
16.05.2013Ada Boralevi Vector Bundles as a tool to study secant varieties 1
23.05.2013??Agnieszka Bodzenta??Vector Bundles... 2
30.05.2013 Corpus Christi holiday
16.05.2013Elisa Postinghel Vector Bundles... 3

Current reading:

Starting March 7th, 2013 we read Geramita's Lecture Notes about zero dimensional subschemes of projective space:
Anthony V. Geramita. "Inverse systems of fat points: Waring's problem, secant varieties of Veronese varieties and parameter spaces for Gorenstein ideals." In The Curves Seminar at Queen's, Vol. X (Kingston, ON, 1995), volume 102 of Queen's Papers in Pure and Appl. Math., pages 2--114. Queen's Univ., Kingston, ON, 1996.
(lecture notes availble upon request) We continue to the end of the academic year with reading about Equations for secant varieties of Veronese and other varieties by Joseph Landsberg and Giorgio Ottaviani.

Past readings:

The first meeting was on Oct 4th. Joachim spoke on the paper of Białynicki-Birula and Schinzel "Representations of multivariate polynomials by sums of univariate polynomials in linear forms".

Starting Oct 11th we were reading the paper by Jessica Sidman and Gregory G. Smith, "Linear determinantal equations for all projective schemes", Algebra and Number Theory or arXiv:0910.2424 and references therein, for instance L. Ein-R. Lazarsfeld, M. Green.

Starting Nov 8th we were reading about Cox rings: Chapter one of this book (ADHL), James McKernan's article "Mori Dream Spaces", this article, this article, and book "Toric varieties" by Cox, Little, Schenck. In the final two meetings, Jarek presented his work with Gavin Brown on maps of toric varieties: arXiv:1004.4924

Further topics:

List of topics suggested so far is here.