Enumerative Geometry 08 Moduli Space

Enumerative Geometry 08 Gromov Witten

Circle $a \cdot x \cdot x + a \cdot y \cdot y + b \cdot z \cdot z + c \cot x \cdot z + d \cdot y \cdot z =0$ Moduli Space $[a:b:c:d] \in \mathbb{P}^3$. Tangency condition resultant.
- Pencil interpolating circle Desmos Calculator
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).
$[a:b:c:d:e:f]$ is $\mathbb{P}^5$ parametrizes conic $a \cdot x \cdot x + b \cdot y \cdot y + c \cdot z \cdot z + d \cot x \cdot z + d \cdot y \cdot z =0$.
Hilbert Scheme parametrizes all closed subschemes.
- Hilbert Scheme of $\mathbb{P}^n$, note $Hilb(\mathbb{P}^n)$ has many snigularities, but its connected components are projective, correspond to closed subschemes with a Hilbert Polynomial (records degree and more).
- $Hilb(\mathbb{P}^2)$ Hilbert polynomial $2 \cdot n +1$ is isomorphic to moduli space of conics: $\mathbb{P}^5$.
- Fano Scheme parametrizes lines in a projective variety.
  - \[F(\mathbb{P}^n)= Gr(2, n+1)\]
  - Counting Lines in projective varieties is counting points in Fano Scheme.
  - Shcemes are varieties with multiplicity points.
  - Chern Class of vector bundles.

Dimension of Moduli Space
- Intersect condition on moduli space gives constants
  - Count conic passing through 5 points: 5 is the dimension of moduli space of conics.
  - Count conic passing through 5 points: 5 is the dimension of moduli space of lines in $p^3$ which is $Gr(2,4)$.
- Incidence Correspondence
Compactify Moduli Space
- Double line could give non-transversal intersection.
- Be careful to choose right compactification.