Algebraic geometry

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Algebraic geometry is the study of solutions of polynomial equations by means of abstract algebra, and in particular ring theory. Algebraic geometry is most easily done over algebraically closed fields, but it can also be done more generally over any field or even over rings. It is not to be confused with analytic geometry, which is use of coordinates to solve geometrical problems.

Affine Algebraic Varieties

One of the first basic objects studied in algebraic geometry is a variety. Let $\mathbb{A}^n$ denote affine $n$-space, i.e. a vector space of dimension $n$ over an algebraically closed field, such as the field $\mathbb{C}$ of complex numbers. (We can think of this as $n$-dimensional "complex Euclidean" space.) Let $R=\mathbb{C}[X_1,\ldots,X_n]$ be the polynomial ring in $n$ variables, and let $I$ be a prime ideal of $R$. Then $V(I)=\{p\in\mathbb{A}^n\mid f(p)=0\mathrm{\ for\ all\ } f\in I\}$ is called an affine algebraic variety.

Projective Varieties

Let k be a field. A projective variety over k is a projective scheme over k. Projective varieties are algebraic varieties. A projective space $\mathbb{P}^n$ is a quotient set with an equivalence class satisfying $(x_0, \dots, x_n) \sim \lambda (x_0, \dots, x_n)$.

Quasiprojective Varieties

The varieties most commonly used, quasiprojective varieties are algebraic varieties given as open subsets of a projective variety with respect to the Zariski topology.

General Algebraic Varieties

Defined in terms of sheafs and patchings.


Let $A$ be a ring and $X=\operatorname{Spec}A$. An affine scheme is a ringed topological space isomorphic to some $(\operatorname{Spec }A,\mathcal{O}_{\operatorname{Spec}A})$. A scheme is a ringed topological space $(X,\mathcal{O}_X)$ admitting an open covering $\{U_i\}_i$ such that $(U_i,\mathcal{O}_{X|U_i})$ is an affine scheme for every $i$.

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