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