---
date: 2023-01-19 15:35
title: Beauville Complex Surfaces
aliases:
  - Beauville Complex Surfaces
annotation-target: 
status: ✅ Started
page_current: 0
page_total: 100000
flashcard: Reading::Surfaces
created: 2023-01-19T15:35
updated: 2024-01-05T21:48
---

Last modified: `=this.file.mday`

# Beauville Complex Surfaces

# Ch. 1: Pic and RR

- [ ] What is the divisor of a section $s\in H^0(\OO_X(D))$ for $D\in \Div(X)^\eff$?
- [x] For $f: X\to X'$ a generically finite degree $d$ morphism, describe $f_* C$ for $C$ an irreducible curve. ✅ 2023-01-25
	- [ ] $f_*(C) = 0$ if $f(C)$ is a point
	- [ ] $f_*(C) = r C'$ if $f(C) = C'$ is a curve where $\ro{f}{C}:C\to C'$ is degree $r$.
- [x] What is the push-pull formula for divisors? ✅ 2023-01-25
	- [ ] $f_* f^* D = \deg(f)\cdot D$.
- [x] What is the local intersection multiplicity $m_x(C \intersect C')$? ✅ 2023-01-25
	- [ ] $m_x(C \intersect C') \da \dim_\CC {\OO_x \over \gens{f, g}}$ where $f,g$ are the equations in $\OO_x$.
- [x] When are $C, C'$ transverse? ✅ 2023-01-25
	- [ ] $m_x(C \intersect C') = 1 \iff \gens{f, g} = \mfm_x$, so $f,g$ form a local coordinate system at $x$.
- [x] What is the ideal sheaf of $C$? ✅ 2023-01-25
	- [ ] $\OO_X(-C)$.
- [x] How is $C . C'$ defined? ✅ 2023-01-25
	- [ ] $C.C' = \sum_{x\in C\intersect C'} \mfm_x(C \intersect C') = h^0(X; \OO_{C\intersect C'})$.
- [x] How is $L.L'$ defined for $L, L'\in \Pic(X)$? ✅ 2023-01-25
	- [ ] $L.L' = \chi(\OO_X) - \chi(L\inv) - \chi((L')\inv) + \chi(L\inv \tensor (L')\inv)$.
- [x] For $C_1, C_2$ curves, what is $\OO_X(C_1) . \OO_X(C_2)$? ✅ 2023-01-25
	- [ ] $\OO_X(C_1) . \OO_X(C_2) = C_1.C_2$
- [x] For $C$ a smooth irreducible curve and $L\in \Pic(X)$, what is $\OO_X(C) . L$? ✅ 2023-01-25
	- [ ] $\OO_X(C) . L = \deg\ro{L}{C}$
- [ ] What is $\deg L$ for $L\in \Pic(X)$?
- [x] For $f:X\to Y$ a generically degree $d$ morphism, what is $f^* D_1 . f^* D_2$? ✅ 2023-01-25
	- [ ] $f^* D_1 . f^* D_2 = \deg(f) (D_1.D_2)$.

# Ch. 2: Birational maps

# Ch. 3: Ruled surfaces