Enumerative Geometry 07 Gromov Witten
Recap
- Chow ring of algebraic variety is rational equivalence of cycle.
- Algebraic construction of Homology allows perturbation.
- The product structure is such that if variety is transverse to another, their product is that of their intersection.
- Could work with scheme (eg: double point).
- Chow ring of \(P^n\) is \(\mathbb{Z}/H^{n+1}\), where \(H\) is the class of a hyperplane.
- The degree of \(dH^r\) is \(d\) and \(r\) is codimension.
- Integral counting points comes from DeRham Cohomology.
- To find the degree, find the degree of intersection with a general hyperplane.
- Veronese injective map degree is same as that of preimage.
- Bidegree
- If map is not injective, mostly work with generally injective: one-to-one on a dense open subset.
Moduli Space
- Parameter Space/variety/scheme/stack for geometric object.
- Projective Space \(P^n_k\) parametrizes line in \(k^{n+1}\).
- Grassmannian \(Gr_k(r,n)\) parametrizes \(r-\)line in \(k^{n+1}\).
- Quarter plane without interior of origin unit circle parametrize triangle (up to similar).
- \(P^5\) parametrizes conic.
- Conic through point desmos calculator
QnA
- Given 3 general circles, count tangent circle.
- Two points in every circle \([1:\pm i:0]\).
- Define \([F:G]\) of interpolation of Pencil \({aF+bG=0}\) by blowup and then strict transform for the two points.
- How many points in intersection of two lines?
- Schubert Calculus studies Chow ring on Grassmannians.
Tomorrow
- 3264 and all that
- Moduli space of curves