This is a “Schubert curve” — related to some work of my own, and to some joint work with Maria Gillespie of Berkeley. To appear, eventually…!
The following situation showed up this spring in my research, and although it ended up not seeming to lead anywhere, I still think there’s something deeper going on.
Consider an algebraic curve defined over . I should emphasize that this is a complex curve with real structure, that is, a Riemann surface with an action of complex conjugation. The fixed points of this action are the curve’s real points.
There are a handful of interesting topological questions we can ask about algebraic curves defined over . For instance:
- Does have any real points at all? Is smooth? What about ?
- How do the real points of sit inside its complex points?
- If is smooth, then it is a disjoint union of circles. How many circles are there? We’ll call this quantity .
This came up in my research recently.
In geometry, “simple nodes” are the simplest, nicest kinds of curve singularities — they have length 1, they’re resolved after a single blowup, they are the only singularities that occur “generically”, and their scheme-theoretic properties seem generally well-understood and well-behaved. (For example, the moduli space of marked points on stable curves of genus exists because of, among other things, the restriction to only nodal curves.)
Up until now, all the Schubert calculus I’ve posted about has been on the Grassmannian . But that isn’t where the story ends. The theory continues to be in development in at least two ways:
- the space of interest might be the flag variety, or more general , for a linear algebraic group and a parabolic or Borel subgroup;
- the ring structure (cohomology theory) might vary: equivariant cohomology, quantum cohomology, K-theory.
So I’ll end this series of posts with a few words on these generalizations: what works, and what — so far — remains mysterious.
When I first planned this mini-course, I expected this topic to be hard to motivate to geometers — why should they care about one algorithm over another for computing the Littlewood-Richardson numbers? But there’s plenty of subtlety in considering the strengths and weakness of the various Littlewood-Richardson rules out there:
- some are fast and convenient for actual computations (on a computer);
- some are “geometric”, that is, they actually describe something happening in ;
- some are “symmetric”, that is, they exhibit directly the symmetries coming from the fact that the Littlewood-Richardson numbers are the structure constants of a commutative, associative ring — and/or they are invariant under transposing partitions;
- some have turned out to generalize more readily to other contexts (such as K-theory).
Last post set up the Schubert decomposition of . We’re going to use it to do some intersection theory computations.
Since Schubert varieties are indexed by partitions, this is really about “multiplying partitions” — and the ring structure will coincide with the structure for multiplying partitions in the contexts of representation theory and symmetric polynomials.
All of these computations are in , the Chow/cohomology ring.
It’s time to set up the Schubert stratification of , a decomposition of the whole variety into locally closed pieces (Schubert cells), each of which is isomorphic to an affine space.
These cells are indexed by partitions and, by general theory, give rise to an additive basis of the Chow, cohomology, and -theory rings of the Grassmannian. And, as a result, intersection theory on has a complete, concrete description in terms of the combinatorics of the Schubert decomposition (i.e. partitions and the Littlewood-Richardson rule).