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# 数学代写|代数几何代写Algebraic Geometry代考|MA66500 Homogenization and dehomogenization

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## 数学代写|代数几何代写Algebraic Geometry代考|Homogenization and dehomogenization

Each polynomial $f \in k\left[x_0, \ldots, x_n\right]$ can be decomposed into homogeneous pieces
$$f=F_0+F_1+\cdots+F_d, \quad d=\operatorname{deg}(f),$$

i.e., each $F_j$ is homogeneous of degree $j$ in $x_0, \ldots, x_n$. An ideal $J \subset k\left[x_0, \ldots, x_n\right]$ is homogeneous if it admits a collection of homogeneous generators. Equivalently, if a polynomial is in a homogenous ideal then each of its homogeneous pieces is in that ideal (see Exercise 9.1).
Dehomogenization with respect to $x_i$ is defined as the homomorphism
\begin{aligned} \mu_i: k\left[x_0, \ldots, x_n\right] & \rightarrow k\left[y_0, \ldots, y_{i-1}, y_{i+1}, \ldots, y_n\right] \ x_i & \rightarrow 1 \ x_j & \rightarrow y_j, \quad j \neq 1 \end{aligned}
For $f \in k\left[y_0, \ldots, y_{i-1}, y_i, \ldots, y_n\right]$, the preimage $\mu_i^{-1}(f)$ contains
$$\left{x_i^D f\left(x_0 / x_i, \ldots, x_{i-1} / x_i, x_{i+1} / x_i, \ldots, x_n / x_i\right): D \geq \operatorname{deg}(f)\right}$$
and equals the affine span of these polynomials. The homogenization of $f$ with respect to $x_i$ is defined
$$F\left(x_0, \ldots, x_n\right):=x_i^{\operatorname{deg}(f)} f\left(x_0 / x_i, \ldots, x_{i-1} / x_i, x_{i+1} / x_i, \ldots, x_n / x_i\right)$$
The homogenization of an ideal $I \subset k\left[y_0, \ldots, y_{i-1}, y_{i+1}, \ldots, y_n\right]$ is the ideal generated by the homogenizations of each $f \in I$.

Given an ideal $I=\left\langle f_1, \ldots, f_r\right\rangle$, the homogenization $J$ need not be generated by the homogenizations of the elements, i.e.,
$$J \neq\left\langle x_i^{\operatorname{deg}\left(f_j\right)} f_j\left(x_0 / x_i, \ldots, x_{i-1} / x_i, x_{i+1} / x_i, \ldots, x_n / x_i\right)\right\rangle_{j=1, \ldots, r}$$
in general.

## 数学代写|代数几何代写Algebraic Geometry代考|Projective varieties

Definition 9.9 A projective variety $X \subset \mathbb{P}^n(k)$ is a subset such that, for each distinguished $U_i \simeq \mathbb{A}^n(k), i=0, \ldots, n$, the intersection $U_i \cap X \subset U_i$ is affine.
Definition 9.10 $\quad X \subset \mathbb{P}^n(k)$ is Zariski closed if $X \cap U_i$ is closed in each distinguished $U_i$. For any subset $S \subset \mathbb{P}^n(k)$, the projective closure $\bar{S} \subset \mathbb{P}^n(k)$ is defined as the smallest closed subset containing $S$.

Definition 9.11 A projective variety $X \subset \mathbb{P}^n(k)$ is reducible if it can be expressed as a union of two closed proper subsets
$$X=X_1 \cup X_2, \quad X_1, X_2 \subsetneq X .$$
It is irreducible if there is no such representation.
We describe a natural way to get large numbers of projective varieties:
Proposition 9.12 Let $F \in k\left[x_0, \ldots, x_n\right]$ be homogeneous of degree $d$. Then there is a projective variety
$$X(F):=\left{\left[a_0, \ldots, a_n\right]: F\left(a_0, \ldots, a_n\right)=0\right} \subset \mathbb{P}^n(k),$$
called the hypersurface defined by $F$. More generally, given a homogeneous ideal $J \subset k\left[x_0, \ldots, x_n\right]$, we define
$$X(J):=\left{\left[a_0, \ldots, a_n\right]: F\left(a_0, \ldots, a_n\right)=0 \text { for each homogeneous } F \in J\right},$$
the projective variety defined by $J$.

## 数学代写|代数几何代写Algebraic Geometry代考|Homogenization and dehomogenization

$$f=F_0+F_1+\cdots+F_d, \quad d=\operatorname{deg}(f),$$

$$\mu_i: k\left[x_0, \ldots, x_n\right] \rightarrow k\left[y_0, \ldots, y_{i-1}, y_{i+1}, \ldots, y_n\right] x_i \quad \rightarrow 1 x_j \rightarrow y_j, \quad j \neq 1$$

〈left 缺少或无法识别的分隔符

$$F\left(x_0, \ldots, x_n\right):=x_i^{\operatorname{deg}(f)} f\left(x_0 / x_i, \ldots, x_{i-1} / x_i, x_{i+1} / x_i, \ldots, x_n / x_i\right)$$

$$J \neq\left\langle x_i^{\operatorname{deg}\left(f_j\right)} f_j\left(x_0 / x_i, \ldots, x_{i-1} / x_i, x_{i+1} / x_i, \ldots, x_n / x_i\right)\right\rangle_{j=1, \ldots, r}$$

## 数学代写|代数几何代写Algebraic Geometry代考|Projective varieties

$$X=X_1 \cup X_2, \quad X_1, X_2 \subsetneq X .$$

〈left 缺少或无法识别的分隔符

\left 缺少或无法识别的分隔符

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