### 数学代考|代数几何代写algebraic geometry代考|Affine and Projective Varieties

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## 数学代考|代数几何代写algebraic geometry代考|Projective and Quasi-Projective Varieties

The $n$-dimensional projective space $\mathbb{P}{\mathbb{C}}^{n}$ is the set of ratios $$\left[x{0}: \cdots: x_{n}\right]$$
of complex numbers. In a ratio, the numbers $x_{0}, \ldots, x_{n}$ are not allowed to all be 0 (although some may be 0 ), and a ratio is considered the same if we multiply all the numbers by the same non-zero number:
$$\left[x_{0}: \cdots: x_{n}\right]=\left[a x_{0}: \cdots: a x_{n}\right]$$
with $a \neq 0 \in \mathbb{C}$.
A projective algebraic set is a set of zeros in $\mathbb{P}{\mathbb{C}}^{n}$ of a set of homogeneous polynomials. (A polynomial is homogeneous if all its monomials have the same degree, which is defined as the sum of exponents of all its variables.) Projective algebraic sets are, by definition, the closed sets in the Zariski topology on $\mathbb{P}{\mathrm{C}}^{n}$. Irreducible projective algebraic sets are called projective varieties. A Zariski open subset (i.e. complement of a Zariski closed subset) in a projective variety is called a quasi-projective variety.One can, for many practical purposes, define an algebraic variety as a quasi-affine or quasi-projective variety. The necessity to always refer to an ambient affine or projective space in such a definition, however, is unsatisfactory, and it is a part of what motivates schemes. However, we must learn about varieties, and some other mathematics, first, before discussing schemes.

## 数学代考|代数几何代写algebraic geometry代考|Regular Functions on a Quasiaffine and Quasiprojective Variety

Let $V$ be a quasiaffine variety (or, more generally, a Zariski open set in an affine algebraic set). A regular function on $V$ at a point $p$ is a function
$$f: U \rightarrow \mathbb{C}$$
where $U$ is a Zariski open set in $V$ with $p \in U$ such that
$$f(x)=\frac{g(x)}{h(x)}$$
where $g(x), h(x)$ are polynomials, and $h(x) \neq 0$ for all $x \in U$ (here we write $x$ for an $n$-tuple: $\left.x=\left(x_{1}, \ldots, x_{n}\right)\right)$.
A regular function on $V$ is a function
$$f: V \rightarrow \mathbb{C}$$
which is regular at every point $p \in V$, i.e. for every $p \in V$, there exists a Zariski open neighborhood $U$ of $p$ such that on $U, f$ is of the form (1.4.1).

A regular function on a quasiprojective variety (or at a point of a quasiprojective variety) is defined the same way as a regular function on a quasiaffine variety with the exception that $g(x), h(x)$ are homogeneous polynomials of equal degree (so that $f(x)$ is well defined on ratios). The definition also applies to Zariski open subsets of projective algebraic sets.

Regular functions on an algebraic variety $X$ form a commutative ring (i.e. we can add and multiply them). This ring is denoted by $\mathbb{C}[X]$. We will now compute the ring of regular functions for varieties of certain kinds.

## 数学代考|代数几何代写algebraic geometry代考|Regular Functions on An

The ring of regular functions on the affine space is simply the ring of polynomials in $n$ variables:
$$\mathbb{C}\left[A_{C}^{n}\right]=\mathbb{C}\left[x_{1}, \ldots, x_{n}\right]$$
To see this, first note that since $A_{C}^{n}$ is irreducible, two polynomials $f, g$ which coincide on a non-empty Zariski open set $U \subseteq \AA_{\mathbb{C}}^{n}$ coincide (since $\mathrm{A}{\mathbb{C}}^{n}=\left(\mathrm{A}{\mathbb{C}}^{n} \backslash U\right) \cup Z(f-g)$ ). Now since $\mathrm{C}\left[x_{1}, \ldots, x_{n}\right]$ has unique factorization (see Theorem $4.1 .3$ below), the same is true for rational functions: Suppose on a non-empty Zariski open set $U \subseteq \mathbb{A}{\mathbb{C}}^{n}$, $$\frac{g{1}}{h_{1}}=\frac{g_{2}}{h_{2}}$$
where $g_{i}, h_{i}$ have greatest common divisor 1 for $i=1,2$, and $h_{i}$ are non-zero on $U$. Then
$$g_{1} h_{2}=g_{2} h_{1},$$
and hence there exists a $u \in \mathbb{C}^{\times}$such that $g_{1}=u g_{2}, h_{1}=u h_{2}$.
Now let $f$ be a regular function on $\mathbb{A}_{\mathbb{C}^{n}}^{n}$. But by what we just observed, in Zariski open neighborhoods of all points, we can write $f=g / h$ with the same polynomials $g, h$ which, moreover, have greatest common divisor 1 . However, if $h \notin \mathbb{C}^{\times}$, by the Nullstellensatz, then, the set of zeros $Z(h)$ of $h$ would be non-empty, so at a point $x \in Z(h)$, we would have a contradiction. Thus, $h \in \mathbb{C}^{\times}$, and $f$ is a polynomial.

## 数学代考|代数几何代写algebraic geometry代考|Projective and Quasi-Projective Varieties

\left[x_{0}: \cdots: x_{n}\right]=\left[a x_{0}: \cdots: a x_{n}\right]

with一种≠0∈C.

F:在→C

F(X)=G(X)H(X)

F:在→C

## 数学代考|代数几何代写algebraic geometry代考|Regular Functions on An

C[一种Cn]=C[X1,…,Xn]

G1H2=G2H1,

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