Lecture notes for the school on birational geometry of moduli spaces in Utah (NOTES)
These notes will be revised frequently in the next few weeks as people point out errors. Please e-mail me any suggestions or corrections.
The Algebraic Geometry Bootcamp will take place at the University of Utah on July 6-10, 2015. For more details see (here)
The Midwest Algebraic Geometry Graduate Conference will take place at UIC on February 22, 23, 2014. For more details see (here)
The Graduate Student Workshop on Moduli Spaces and Bridgeland Stability will take place at UIC on March 23-25, 2013. For more details see ( Stability Workshop )
The Istanbul Summer School in Algebraic Geometry will take place June 17-21, 2013 at the Istanbul Mathematical Sciences Center. For more information see ( Istanbul )
Teaching:
This year I will be teaching Algebra Math 330, Graduate Algebra I Math 516 in the fall and Undergraduate Algebraic Geometry Math 494 in the spring. Last year I taught Topics in Algebraic Geometry MATH 571 and Commutative and Homological Algebra MATH 520.
I am on the editorial board of Advances in Geometry. I served on the editorial board of the Central European Journal of Mathematics (2009--2014) until the decisions of the publisher led most of the editorial board to resign.
XXIII National Mathematics Symposium, Kayseri Turkey, August 2010.
Lecture 1, Lecture2
Papers and preprints:
Comments, corrections and
suggestions are always welcome.
The effective cone of the moduli space of sheaves on the plane (joint with Jack Huizenga and Matthew Woolf) (pdf) Preliminary version, comments and corrections welcome.
Eckardt loci on hypersurfaces (joint with Artie Prendergast-Smith) (pdf) Preliminary version, comments and corrections are welcome.
Interpolation, Bridgeland stability and monomial schemes in the plane (joint with Jack Huizenga) (pdf) To appear in Journal de Mathematiques pures et appliquees.
Extremal effective divisors on the moduli space of n-pointed genus one curves (joint with Dawei Chen) (pdf) To appear in Mathematische Annalen.
Flexibility of Schubert classes (joint with Colleen Robles) (pdf) Differential Geometry and its applications, 31 no. 6 (2013), 759--774.
Fano manifolds of index n-1 and the cone conjecture (joint with Artie Prendegast-Smith) (pdf) To appear in International Mathematics Research Notices.
Symplectic retriction varieties and geometric branching rules II. Revised version: (pdf) old file: (pdf) To appear in Journal of Combinatorial Theory Series A.
The birational geometry of the Hilbert scheme of points on surfaces. (pdf) (joint with Aaron Bertram). Birational geometry, rational curves and arithmetic, Simons Symposia, Springer 2013, 15--55.
The minimal model program for the Hilbert scheme of points on the plane and Bridgeland stability. (pdf) (joint with Daniele Arcara, Aaron Bertram and Jack Huizenga) Advances in Mathematics, volume 235 (2013), 580--626.
Rigidity of Schubert classes in orthogonal Grassmannians. (pdf) To appear in Israel Journal of Mathematics.
Surfaces of low degree containing a canonical curve. (pdf) Computational Algebraic and Analytic Geometry, Contemporary Mathematics, 572, 2012.
Parameter spaces of Schubert varieties in hyperplane sections of Grassmannians (with my student Richard Abdelkerim, this is part of his thesis) (pdf) Journal of Pure and Applied Algebra
vol. 216 (2012), 800--810.
Singularities of generalized Richardson varieties (joint with Sara Billey) (pdf ). Communications in Algebra. 40:4, 1466--1495.
Rigid and non-smoothable Schubert classes (pdf ). Journal of Differential Geometry vol. 87, no. 3 (2011), 493--514.
Towards Mori's program for the moduli space of stable maps (joint with Dawei Chen, with an appendix by Charley Crissman) (pdf). American Journal of Mathematics vol. 133 no.5 (2011), 1389--1419.
Stable base locus decompositions of Kontsevich moduli spaces (joint with Dawei Chen) (pdf). Michigan Mathematical Journal, vol. 59, no. 2 (2010), 435--466.
Hilbert scheme of a pair of codimension two linear subspaces (joint with Dawei Chen and Scott Nollet) (pdf). Communications in Algebra vol. 39 no.8 (2011), 3021--3043.
Rational curves on smooth cubic hypersurfaces (joint with Jason Starr) (pdf). International Mathematics Research Notices Article RNP102 (2009), 16 pages, doi:10.1093/imrn/rnp102.
Restriction varieties and geometric branching rules. (pdf). Advances in Mathematics. vol. 228 no. 4 (2011), 2441--2502.
The quantum cohomology of flag varieties and the periodicity of the Schubert structure constants. (pdf) Mathematische Annalen vol. 346 no. 2 (2010), 419--447.
A Littlewood-Richardson rule for partial flag varieties. (pdf). Currently under revision. This is a preliminary version of a Littlewood-Richardson rule for arbitrary partial flag varieties. Any comments, corrections and suggestions are welcome. GALLERY (html)
A Littlewood-Richardson rule for two-step flag varieties (pdf). For examples see Gallery (html). Old version is available here: (pdf).
Inventiones Mathematicae volume 176, no 2 (2009) p. 325--395 DOI: 10.1007/s00222-008-0165-3
Geometric positivity in the cohomology of homogeneous
spaces and generalized Schubert calculus (joint with Ravi Vakil) (pdf). Algebraic Geometry, Seattle 2005 volume 1, p. 77--124.
The cohomology of the space of k-planes on
quadrics. Coming soon
Divisors on the space of maps to Grassmannians (joint with
Jason Starr). (pdf). International Mathematics Research Notices, vol. 2006, Article ID 35273, 25 pages, 2006.
The effective cone of the Kontsevich moduli space (joint
with Joe Harris and Jason Starr). (pdf). Canadian Mathematical Bulletin. vol. 51 no. 4 (2008), pages 519-534.
The ample cone of the Kontsevich moduli space (joint with Joe
Harris and Jason Starr) (pdf).
Canadian Journal of Mathematics. vol. 61 no. 1 (2009), pages 109-123.
Gromov-Witten invariants of jumping curves (pdf). Transactions of the American Mathematical Society. 360 (2008), p. 989-1004.
Degenerations of surface scrolls and Gromov-Witten invariants of
Grassmannians (pdf).
Journal of Algebraic Geometry. 15 (2006), p. 223-284.
The enumerative geometry of Del Pezzo surfaces via degenerations
(pdf). American
Journal of Mathematics. vol. 128 no. 3 (2006), p. 751-786.
The arithmetic and the geometry of Kobayashi hyperbolicity (pdf). Snowbird lectures in algebraic geometry. Contemporary Mathematics, vol 388 (2005) p. 77-88.
A first glimpse at the minimal model program (pdf). (with 8 coauthors) Snowbird lectures in algebraic geometry. Contemporary
Mathematics, vol 388 (2005) p. 17-42.
Degenerations of scrolls and Del Pezzo surfaces and applications
to enumerative geometry, Thesis, Harvard University, 2004.