Quick summarized of algebraic geometry

1月 25, 2021

Summary 1:

一開始就是先講spec and proj 的構造

Spec versus Proj constructions:

Summary 2:

這部分不簡單~~

Functoriality:

Summary 3:

Closed subschemes:

這部分是generalized 從原來的ring 構造出的spectrum 變成從sheaf 構造

這可能是代數幾何的精華所在之一

Affine cone and Projective Cone

Summary 4:

Affine versus projective cones: (Different references use slightly different names for these.)

Summary 5:

Morphisms to affine/projective cones: Let $Y$ be a scheme, $E$ be a quasi-coherent $O_{Y}$ -module, and $q : X \to Y$ be a $Y$ -scheme.

Summary 6:

Segre embedding:

Summary 7:

這也是很神奇的主題 invertible sheaf 在代數幾何裡面特別是projecive space 上

有非常重要的計算

Very ample invertible sheaves:

Summary 8:

Ample invertible sheaves: Let $X$ be a scheme and $L$ be an invertible $O_{X}$ -module. Roughly speaking, ampleness of $L$ means that high powers of $L$ have plenty of global sections. We give several equivalent conditions for ampleness. Three of them are as follows: