Ibadan Lectures on Toric Varieties

Toric varieties are perhaps the most accessible class of algebraic varieties. They often arise as varieties parameterized by monomials, and their structure may be completely understood through objects from geometric combinatorics. While accessible and understandable, the class of toric varieties is also rich enough to illustrate many properties of algebraic varieties. Toric varieties are also ubiquitous in applications of mathematics, from tensors to statistical models to geometric modeling to solving systems of equations, and they are important to other branches of mathematics such as geometric combinatorics and tropical geometry. These notes are based on, and significantly extend, Frank Sottile's short course of four lectures at the CIMPA school on Combinatorial and Computational Algebraic Geometry in Ibadan, Nigeria 12--23 June 2017.Comment: 40 pages, many figure

The special Schubert calculus is real

We show that the Schubert calculus of enumerative geometry is real, for special Schubert conditions. That is, for any such enumerative problem, there exist real conditions for which all the a priori complex solutions are real.Comment: 5 page

Pieri-type formulas for maximal isotropic Grassmannians via triple intersections

We give an elementary proof of the Pieri-type formula in the cohomology of a Grassmannian of maximal isotropic subspaces of an odd orthogonal or symplectic vector space. This proof proceeds by explicitly computing a triple intersection of Schubert varieties. The decisive step is an explicit description of the intersection of two Schubert varieties, from which the multiplicities (which are powers of 2) in the Pieri-type formula are deduced.Comment: LaTeX 2e, 24 pages (9 pages is an appendix detailing the proof in the symplectic case). Expanded version of MSRI preprint 1997-06

Real Rational Curves in Grassmannians

Fulton asked how many solutions to a problem of enumerative geometry can be real, when that problem is one of counting geometric figures of some kind having specified position with respect to some general fixed figures. For the problem of plane conics tangent to five general conics, the (surprising) answer is that all 3264 may be real. Similarly, given any problem of enumerating p-planes incident on some general fixed subspaces, there are real fixed subspaces such that each of the (finitely many) incident p-planes are real. We show that the problem of enumerating parameterized rational curves in a Grassmannian satisfying simple (codimension 1) conditions may have all of its solutions be real.Comment: 9 pages, 1 eps figure, uses epsf.sty. Below the LaTeX source is a MAPLE V.5 file which computes an example in the paper, and its outpu