Questions in category: 代数几何 (Algebraic Geometry)

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## 1. [Def] Scheme 的定义

Posted by haifeng on 2023-05-04 11:26:36 last update 2023-05-04 11:26:36 | Answers (0) | 收藏

A scheme is a ringed space $X,\mathcal{O}_X$ for which every point has a neighbourhood $U$ such that the ringed space $U,\mathcal{O}_{X}|_U$ is isomorphic to $\mathrm{Spec}A$, where $A$ is some ring.

References:

[1] Shafarevich, Basic Algebraic Geometry 2.

## 2. 周炜良定理(Chow's theorem)

Posted by haifeng on 2022-01-19 10:34:26 last update 2022-01-19 10:57:58 | Answers (0) | 收藏

Chow 定理.   射影空间 $P_n(\mathbb{C})$ 中紧致解析簇是代数的.

References:

[1] Wei-Liang Chow, On compact complex analytic varieties, Amer. J. Math. 71 (1949), 893–914. MR 33093

[2] R. O. Wells, Differential Analysis on Complex Manifolds, GTM 65.

## 3. Riemann-Roch 定理

Posted by haifeng on 2021-11-30 17:13:24 last update 2022-01-26 15:43:10 | Answers (0) | 收藏

$\ell(D)-\ell(K_C-D)=\deg(D)-g(C)+1,$

$\ell(D)=\dim_k H^0(C,\mathcal{L}(C))$

References:

## 4. Congruence Subgroups

Posted by haifeng on 2021-10-14 22:48:49 last update 2021-10-14 22:55:53 | Answers (0) | 收藏

## Congruence Subgroups

$\Gamma(N)=\{\gamma\in SL_2(\mathbb{Z})\mid \gamma\equiv 1_2\pmod N\}$

$\Gamma(N)=\biggl\{ \begin{bmatrix} a & b\\ c & d\end{bmatrix}\in SL_2(\mathbb{Z})\biggl|\ a\equiv d\equiv 1,\ b\equiv c\equiv 0\ \mod N\mathbb{Z} \biggr\}$

Congruence Subgroups 是指 $SL_2(\mathbb{Z})$ 中包含 $\Gamma(N)$ 的任一子群, $N$ 是某个自然数.

Math 252: Congruence Subgroups (wstein.org)

https://wstein.org/edu/Fall2003/252/lectures/09-19-03/index.html

## 5. Shimura varieties

Posted by haifeng on 2021-06-18 11:14:09 last update 2021-06-18 11:14:09 | Answers (0) | 收藏

Shimura varieties

http://conference.bicmr.pku.edu.cn/meeting/index?id=95

## 6. 代数与几何的联系

Posted by haifeng on 2021-03-29 14:40:26 last update 2021-03-29 15:30:55 | Answers (0) | 收藏

代数 几何

$V: J\mapsto V(J)$ $k[x_1,\ldots,x_n]$ 中的理想 $J$ $\mathbb{A}_k^n$ 中的代数集 $V(J)$ $V$ 是满射, 但不是单射
$I(X)\leftarrow X:\ I$ $I(X)$ $X\subset\mathbb{A}_k^n$ $I$ 既非单射亦非满射

$V(T):=\{P\in\mathbb{A}_k^n\mid\text{对所有}\ f\in T, \text{有}\ f(P)=0\}.$

$\begin{array}[rcl] V:\ \{A\text{的理想}\}&\rightarrow&\{\mathbb{A}_k^n\text{中的代数集}\}\\ J&\mapsto& V(J) \end{array}$

$I(X):=\{f\in A\mid f(P)=0\ \forall\ P\in X\}$

$\begin{array}[rcl] I:\ \{\mathbb{A}_k^n\text{的子集}\}&\rightarrow&\{A\text{中的理想}\}\\ X&\mapsto& I(X) \end{array}$

Claim 1. $V$ 不是单射.

References:

Klaus Hulek 著, 《初等代数几何》

## 7. 关于复数多项式的平方式的一个引理

Posted by haifeng on 2021-01-05 11:33:08 last update 2021-01-05 11:33:08 | Answers (1) | 收藏

References:

Klaus Hulek 著 《初等代数几何》P.8

## 8. 尼尔(Neil)抛物线

Posted by haifeng on 2021-01-05 09:57:50 last update 2021-01-05 10:27:00 | Answers (0) | 收藏

$C:\quad y^2=x^3.$

$\begin{eqnarray} \varphi:\ \mathbb{R}&\rightarrow&\mathbb{R}^2\\ t&\mapsto &(t^2,t^3) \end{eqnarray}$

$\mathrm{d}\varphi=(\mathrm{d}\varphi_1,\mathrm{d}\varphi_2)=(2t\mathrm{d}t,3t^2\mathrm{d}t)$

References:

Klaus Hulek  著 《初等代数几何》P.6

## 9. 不能被有理参数化的实曲线

Posted by haifeng on 2020-12-17 08:08:00 last update 2021-03-09 07:12:23 | Answers (1) | 收藏

$C_{\lambda}:\quad y^2=x(x-1)(x-\lambda),\qquad(\lambda\in\mathbb{R})$

$g^2=f(f-1)(f-\lambda),\quad (\lambda\neq 0,1)$

$g^2=f(f-1)(f-\lambda),\quad (\lambda\neq 0,1)$

$(f,g):\quad k\rightarrow C_{\lambda},\quad (f,g\in k(t))$

$C_0:\ y^2=x^3-x^2$

$C_1:\ y^2=x(x-1)^2$

(1)

$y^2=x^3+x^2$ 具有参数化: $t\mapsto(t^2-1, t^3-t)$. (参见问题1500)

$\begin{eqnarray} \varphi:\ \mathbb{R}&\rightarrow &\mathbb{R}\\ t&\mapsto&(t^2+1, t^3+t) \end{eqnarray}$

(2) 对于 $C_1$, 令 $u=x-1$, 则 $C_1$ 变为(右移为) $y^2=(u+1)u^2=u^3+u^2$, 此即问题1500.

$\begin{eqnarray} \varphi:\ \mathbb{R}&\rightarrow &\mathbb{R}\\ t&\mapsto&(t^2, t^3-t) \end{eqnarray}$

Question:

Remark:

$y^2=x(x-A)(x+B)$

References:

[1] Klaus Hulek 著, 胥鸣伟 译 《初等代数几何》 高等教育出版社.

[2] Richard K. Guy, Unsolved Problems in Number Theory, 科学出版社.

## 10. 证明等式 $x^4+y^4=1$ 只有有理解 $(\pm 1,0)$ 和 $(0,\pm 1)$.

Posted by haifeng on 2020-11-30 16:45:42 last update 2021-03-09 07:14:11 | Answers (0) | 收藏

[Hint] 使用无穷递降法证明 $x^4+y^4=z^2$ 无正整数解.

References:

Klaus Hulek 著 《初等代数几何》,  胥鸣伟 译.   P.16

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