PRAGMATIC 2010: Mori theory in a variety of flavors
Paltin Ionescu and Jaroslaw Wisniewski
Introduction: Thirty years ago Shigefumi Mori published two
ground breaking papers giving birth to a theory now bearing his
name. In the first of these papers, devoted to a proof of the
Frankel-Hartshorne conjecture, Mori proved that Fano manifolds are
covered by rational curves. In the second paper, classical facts about
the birational geometry of surfaces were extended to threefolds whose
canonical divisor is not nef. To this end, Mori used rational curves
to describe morphisms of such varieties and related them to the cone
of effective one cycles. These concepts were subsequently used in the
Minimal Model Program (MMP) aimed at birational classification of
complex projective varieties.
Apart from its vital contribution to the
success of MMP, Mori theory had enormous impact on biregular geometry
and it changed geometer's view on many classical problems. The
present course will be devoted to this latter field of applications of
Mori theory. The aim of the course is to get the participants
acquainted with basic tools of Mori theory and their applications in
dealing with problems related to biregular geometry of projective
manifolds.
Topics: outline of the main problems.
- Problems proposed by Paltin Ionescu:
- Manifolds covered by lines; properties inherited by the variety of
lines through a general point.
- Applications to dual defective and special secant defective
manifolds (varieties with quadratic entry locus). Extremality
properties. Projections.
- Quadratic manifolds; classification problems in small codimension.
- Comparing Kleiman--Mori cones of a variety and an ample
divisor. The case of P^1 bundles.
- Special covering families of rational curves with applications to
rationality and unirationality.
- Problems proposed by J. Wisniewski:
- Cox rings of Du Val and symplectic singularities and their GIT
quotients.
- Structure of the movable cone of blow-ups of the projective
space, relation to Cremona group.
Program:
- Schedule for the first week: lectures in the morning: 9:30-13:30,
1h30min=2x45min by each of the main teachers with breaks; group
tutorials in the afternoon: 15:30-18:30.
Readings:
- General texts: introduction to Mori Theory and rational curves
- Koll'ar, Rational curves on algebraic varieties, Springer EMG, 1996.
- Debarre, Higher-dimensional algebraic geometry, Springer UTX, 2001.
- Matsuki, Introduction to the Mori program, Springer UTX, 2002.
- Koll'ar, Mori, Birational geometry of algebraic varieties,
Cambridge Tracts in Math, 1998.
- General texts: projective geometry: secants, tangents, dual varieties,
projections, second fundamental form.
- Zak, Tangents and Secants of Algebraic Varieties,
Transl. Math. Monogr. vol. 127, AMS 1993.
- Russo, Tangents and Secants of Algebraic Varieties, Publicacoes
Matematicas, IMPA, 2003.
-
Russo,
Geometry of Special Varieties, notes at author's web page.
- General texts: toric varieties and GIT
- Cox,
What is a toric variety? a crash course on toric varieties, more
available at
http://www3.amherst.edu/~dacox/
- Cox, Little, Schenck, Toric
varieties, chapters 1-7, more available
at
http://www.cs.amherst.edu/~dac/toric.html .
- Fulton, Introduction to toric varieties. Annals of Mathematics
Studies 131, Princeton 1993.
- Dolgachev, Lectures on invariant theory, LMS , Cambridge 2003.
Ch. 6-8 copied here.
- Reading list of Paltin Ionescu: Mori theory for subvarieties of
the projective space.
- Russo, Varieties with quadratic entry locus, I,
arXiv:math/0701889, or
Math. Ann., 344 (2009) 597--617.
- Ionescu, Russo, Varieties with quadratic entry locus, II, arXiv:math/0703531, or
Compositio Math. 144 (2008) 949--962
- Ionescu, Russo, Conic-connected Manifolds,
arXiv:math/0701885,
or J. Reine Angew. Math. 2010.
- Ionescu, F. Russo, Manifolds covered by lines, defective
manifolds and a restricted Hartshorne Conjecture,
arXiv:0909.2763
- Ionescu, On manifolds of small
degree, Comment. Math. Helv. 83 (2008) 927--940.
- Beltrametti, Ionescu, On manifolds swept
out by high dimensional quadrics, Math. Zeit. 260 (2008) 229--234;
Corrigendum
- Russo, Lines on projective varieties and applications, soon to be
on arXiv.
- Beltrametti, Ionescu, A view on extending morphisms from ample
divisors, arXiv:0907.2338,
or in Contemporary Math. 496 (2009) 71--110.
- Ionescu, Birational Geometry of Rationally Connected Manifolds
via Quasi-lines,
arXiv:math/0502160, or in "Projective Varieties with Unexpected
Properties", Siena 2004, de Gruyter (2005) 317--335.
- Lopez, Ran, On the irreducibility of secant cones, and an
application to linear normality,
arXiv:math/0111147 or
Duke Math. J. 117 (2003) 389--401.
- Wisniewski, Uniform vector bundles on Fano
manifolds and an algebraic proof of Hwang-Mok characterization of
Grassmannians, or find it in Complex geometry dedicated to Hans
Grauert, see it in
google books.
- Reading list of J. Wisniewski: Mori Dream Spaces, quotients, and
blow-ups of projective spaces
-
Reid,
What is a flip? scanned lecture notes from author's web page.
-
Reid,
The Du Val singularities A_n, D_n, E_6, E_7, E_8, from author's
web page.
- Hu, Keel, Mori Dream
Spaces and GIT
from http://arxiv.org/abs/math/0004017
or Michigan Math. J. 48 (2000), 331--348.
- Dolgachev, Hu,
Variation of geometric invariant theory quotients
from http://arxiv.org/abs/alg-geom/9704022
or Inst. Haut. Sci. Publ. Math. No. 87 (1998), 5--56.
- Laface, Velasco, A
survey on Cox rings, from
http://arxiv.org/abs/0810.3730.
- Dolgachev, Weyl groups and Cremona transformations.
Singularities, Part 1 (Arcata, Calif., 1981), 283--294,
Proc. Sympos. Pure Math., 40, Amer. Math. Soc., Providence, RI, 1983.
here
at google books and a scan here
-
Wisniewski,
Toric Mori theory and Fano manifolds, in Geometry of toric
varieties, 249--272, Semin. Congr., 6, Soc. Math. France, Paris, 2002.
-
Mukai,
Finite generation of the Nagata invariant rings in A-D-E cases, RIMS preprint 2005.
- Castravet,
Tevelev, Hilbert's 14th
problem and Cox rings,
from http://arxiv.org/abs/math/0505337
or Compos. Math. 142 (2006), no. 6, 1479--1498.
-
Dolgachev,
Reflection groups in algebraic geometry, Bull. AMS 45 (2008),
1-60.
- Tauvel, Yu, Lie algebras and algebraic groups. Springer
2005 Root systems, Chapter
18, a copy here.
- Java models: examples for JW lectures
- Slides: part 1
and part 2
Contact: question and comment regarding this web page send to
J.Wisniewski[at]mimuw.edu.pl.