---
date: 2023-01-14 20:42
title: Kodaira Dimension Notes
aliases:
  - Kodaira Dimension Notes
annotation-target: https://people.math.harvard.edu/~mpopa/483-3/notes.pdf
flashcard: Research::Definitions
created: 2023-03-30T13:04
updated: 2024-01-29T22:12
---

# Kodaira Dimension Notes

- [x] How are plurigenera related to the arithmetic or geometric genus? ✅ 2023-01-14
	- [ ] $P_1(X) = g(X)$ for smooth projective curves.
- [x] What is $P_m(X)$ for $X$ a smooth complete curve, in terms of $g(X)$? ✅ 2023-01-14
	- [ ] $g=0 \implies X \cong \PP^1 \implies \omega_X \cong \OO_{\PP^1}(-2) \implies P_{\geq 0}(X) = 0$.
	- [ ] $g=1 \implies \omega_X\cong \OO_X \implies P_{\geq 0}(X) = 1$.
	- [ ] $g\geq 2 \implies \deg \omega_X^m = m(2g-2)\implies P_{\geq 2}(X) = m(2g-2) - (g-1)$ 
- [x] What is $P_m(X)$ for $X \subseteq \PP^n$ a smooth hypersurface of degree $d$? ✅ 2023-01-14
	- [ ] Write $\OO_X(1) = \ro{\OO_{\PP^n}(1)}X$, then $\omega_X \cong \OO_X(d - (n+1) )$.
	- [ ] $d < n + 1 \implies P_{\geq 0}(X) = 0$
	- [ ] $d = n +1 \implies \omega_X \cong \OO_X \implies P_{\geq 1}(X) = 1$
	- [ ] $d > n+1 \implies \omega_X \in \Pic(X)^{\vamp} \implies \cdots$ $$P_{m \gg 0}(X) = {d(d-n-1) \over(n-1)! }\cdot m^{n-1} + \bigo(m^{n-2})$$
- [x] What is $\omega_X$ for $X \subseteq \PP^{n+1}$ a smooth complete intersection of hypersurfaces of degrees $d_1,\cdots, d_k$? ✅ 2023-01-14
	- [ ] $\omega_X \cong \OO_X(\sum d_i - (n+1))$.
- [x] What is a weak CY? ✅ 2024-01-29
	- [ ] $X$ smooth projective with $\omega_X \cong \OO_X$.
- [x] What is a (strong) CY? ✅ 2023-01-14
	- [ ] A weak CY with $H^i(\OO_X) = 0$ for $0 \lt i \lt \dim(X)$ and $\pi_1(X(\CC)) = 1$.
- [x] Give an example of a weak but not strong CY. ✅ 2023-01-14
	- [ ] An abelian variety.
- [x] Give examples of strong CYs. ✅ 2023-01-14
	- [ ] K3s, since $\omega_X\cong \OO_X$ and $H^1(\OO_X) = 0$, e.g. 
		- [ ] A quartic in $\PP^3$ (hypersurface of degree 4)
		- [ ] A complete intersection of type $(2, 3)$ in $\PP^4$.
		- [ ] A complete intersection of type $(2,2,2$) in $\PP^5$.
	- [ ] Hypersurfaces of degree $n+1$ in $\PP^n$.
- [x] Classify hypersurfaces $X\subseteq \PP^n$ by Kodaira dimension. ✅ 2023-01-14
	- [ ] $\kappa(X) = -\infty \iff d\leq n$.
	- [ ] $\kappa(X) = 0 \iff d= n$.
	- [ ] $\kappa(X) = \dim(X) \iff d\geq n+2$.
- [x] What is the fundamental SES for a divisor $D$ on a surface $X$? ✅ 2023-01-25
	- [ ] $\OO_X(-D) \injects \OO_X \surjects \OO_D$.
- [x] How is $C.D$ defined for curves on a surface? ✅ 2023-01-25
	- [ ] $$C.D = \sum_{p\in C\intersect D}(C.D)_p, \quad (C.D)_p \da \dim_k \OO_{X, p}/\gens{f, g}$$ where $C \transverse D \iff \mfm_x = \gens{f, g}$ for $f,g$ their local equations.
- [x] How is $D\in \Div(X)$ related to normal bundles? ✅ 2023-01-25
	- [ ] $D^2 = \deg \mcn_{D/X}$.
- [x] What is $C.D$ for $C, D \subseteq \PP^2$? ✅ 2023-01-25
	- [ ] $C\sim m H$ and $D\sim n H$ for $H$ a hyperplane (here a line), so $$C.D = (mH).(nH) = mnH^2 = mn$$ where $H^2 = (H.H') = \size(H\intersect H') =1$ by moving $H$ to $H'$.
- [x] What is the intersection matrix for $\PP^1\times \PP^1$? ✅ 2023-01-25
	- [ ] Write $\Pic(\PP^1\times \PP^1) = \gens{f_1, f_2}_\ZZ$ for $f_i$ the class of a fiber to get $$\matt{0}{1}{1}{0},\qquad \text{i.e. }f_1 .f_2 = 1,\quad f_1^2 = f_2^2 = 0$$
- [x] What is the genus formula for a curve on a smooth surface? ✅ 2023-01-25
	- [ ] $$\omega_C \simeq \ro{\qty{\omega_X \otimes \mathcal{O}_X(C)}}{ C} \underset{\deg}\implies 2g-2=\left(C+K_X\right) \cdot C$$
- [x] What is RR for surfaces? ✅ 2023-01-25
	- [ ] For $D\in \Div(X)$, $$\chi(\OO_X(D)) - \chi(\OO_X) = {D\cdot (D-K_X) \over 2}$$