MA665: Toric varieties: an introduction to algebraic geometry

Final project - rules.

There will be two type of projects to be prepared: presentation (P)
and research (R). Each presentation project is to be performed by a
single person who will prepare notes (2-4 pp) and a presentation
(40-60 min). The presentations regarding two-part subjects (1 and 1a,
2 and 2a below) should be coordinated (in particular the person
preparing the second part may and should rely on the contents of the
first part). Each research project can be done by a team (1-3 person)
which will prepare a report (2-4pp) and a presentation (40-60
min). Each presentation will be discussed in the class.

Timetable:

* announcing complete list of possible subjects for
presentation/research projects: by Feb 11th, 12-noon, at
http://www.mimuw.edu.pl/~jarekw/toric/

* submitting proposals regarding projects: from Feb. 24th 12-noon to
Feb 26th 12-noon, by sending an e-mail to wisniew@math.purdue.edu

Each person enlisting for presentation project should provide three
titles at least, in the order of preference - assigning of
presentations will be done according to the order of preference and
arrival of the proposals. People enlisting for a research project are
expected to collaborate and write a joint report and prepare a joint
presentation.

* announcing scheduled projects: Feb. 28th, by 5pm at
http://www.mimuw.edu.pl/~jarekw/toric/

* submitting notes and reports regarding projects: April 1st during the class
12:00-1:15

* announcing the schedule of presentations: April 2nd, by 5pm at
http://www.mimuw.edu.pl/~jarekw/toric/

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Proposals for projects.

=================

Presentations.

0. The magic number twelve for toric surfaces. 

   Readings: start with lecture 7 from Grenoble series

1. 1-cycles on toric varieties and toric cone theorem.
   
   Readings: Reid, Decomposition of toric morphisms, in Birkhauser
   Progress in Math36, MR0717617 (85e:14071)

   Matsuki, Introduction to Mori program, Springer [section 14.1]

   Wisniewski, Toric Mori Theory and Fano manifolds, 
   http://www.mimuw.edu.pl/~jarekw/postscript/toric.ps	 

1a.Toric contraction theorem.
   Readings: as above [Reid], [Matsuki 14.2], [Wisniewski].

2. Toric Fano manifolds. Define Fano polytopes, explain low
   dimensional cases. Show the contractions.
   
   Readings: start with Batyrev, On the classification of toric Fano
   $4$-folds. Algebraic geometry, 9.  J. Math. Sci. (New York) 94
   (1999), no. 1, 1021--1050.  MR1703904 (2000e:14088) and use
   references there

2a. Boundness of toric Fano manifolds. Task: why there's only a finite
    number of Fano polytopes (in each dimension)?

    Readings: as above


3.  Euler sequence for toric varieties. Task: define Euler sequence.

    Batyrev, Mel'nikov, A theorem on nonextendability of toric
    varieties. (Russian) Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1986,
    no. 3, 20--24, 118.  MR0848335 (87j:14078) English translation:
    Moscow Univ. Math. Bull.  41 (1986), no. 3, 23--27.

    Jaczewski, Generalized Euler sequence and toric varieties.
    Classification of algebraic varieties (L'Aquila, 1992), 227--247,
    Contemp. Math., 162, MR1272701 (95d:14054)


Research.

1. Toric varieties with orbifold contact structure.  Question:
   describe Q-factorial toric varieties (or orbifolds) which admit
   contact structure. Possible additional assumptions: isolated
   singularities, low dimension (3 & 5).
   
   Readings:
      
   To understand orbifolds: Blache, Chern classes and
   Hirzebruch-Riemann-Roch theorem for coherent sheaves on
   complex-projective orbifolds with isolated singularities. Math. Z.
   222 (1996), 7--57 MR1388002 (97d:14015)

   To understand the smooth result: Druel, Structures de contact sur
   les variétés toriques. (French) [Contact structures on toric
   varieties] Math. Ann.  313 (1999), no. 3, 429--435. MR1678533
   (99m:14099)

   Why contact structures are important: Lebrun, Fano manifolds,
   contact structures and quaternionic geometry, Int. journ. of Math.
   6, 419--437, 1995 MR1327157 (96c:53108)
   

2. Del Pezzo polytopes/cones and toric varieties. The problem has
   several layers, below I explain only the simplest one. We define
   inductively two classes of dual cones, which we will call C and D
   del Pezzo cones.  Namely, let C and D be dual cones (rational
   polyhedral strongly convex of maximal dimension) in dual vector
   spaces of dimension r.  We mark the rays (1-dimensional faces) of
   the cone D by letters (a) and (b). Now we set the inductive
   conditions: (0) We assume that in dimension 2 the cone D can have
   markings (a,a) and (a,b) while in dimension the cone D is
   simplicial and has markings (a,a,b) only.  (1) All faces of C of
   dimension < n-1 are simplicial and n-1 dimensional faces of C are
   either simplicial (we call them type (b)) or octahedral (call them
   type (a)) which means generated by 2(n-1) vertices satisfying the
   relation u_1+v_1=u_2+v_2=....=u_{n-1}+v_{n-1}; we mark the rays of
   D by (a) and (b), respectively, and assume that the faces of D of
   dimension bigger than or equal to 2 are of the same type (as cones
   with marked rays).

Problem: prove that n is 8 at most and C and D are defined uniquely
(in the sense of combinatorics??). Find the number of rays in C and
compare it with the classical results (see [Manin] below). Motivation:
this construction is coming from studying cohomlogy of del Pezzo
surfaces. Extension: these cones live in spaces with lattices and the
cone C can be used to define Fano manifolds with special singularities
and lots of symmetries, which perhaps deserve to be studied.


    Readings: 

    about del Pezzo surfaces: Manin, Cubic forms: algebra,
    geometry, arithmetic, 197?, MR0360592 (50 #13040)

    partial answers: Stalij, Warsaw's MSc Thesis at
    http://www.mimuw.edu.pl/~jarekw/postscript/marcinst.ps

    why symmetric Fano's may be interesting: Batyrev, Selivanova,
    Einstein-Kähler metrics on symmetric toric Fano manifolds.
    J. Reine Angew. Math.  512 (1999) MR1703080 (2000j:32038)