Fulton 1.2

Posted on November 15, 2017

1.8.* Show that the algebraic subsets of \mathbb{A}^1(k) are just the finite subsets, together with \mathbb{A}^1(k) itself.

Suppose A=V(S) for some S\subset k[X]. If S=\emptyset then A= \mathbb{A}^1(k). Otherwise let F\in S. Note A\subset V(F). Since F has finitely many roots, V(F) is finite. Thus A is finite.

Conversely, by (4) any finite subset of \mathbb{A}^1(k) is algebraic.

1.9. If k is a finite field, show that every subset of \mathbb{A}^n (k) is algebraic.

Since k is finite, \mathbb{A}^n(k) is finite. Thus every subset of \mathbb{A}^n(k) is finite and therefore algebraic by (4).

1.10. Give an example of a countable collection of algebraic sets whose union is not algebraic.

Let F_n=(x-n)^2\in\mathbb{R}[X], n\in\mathbb{Z}. We claim S=\bigcup_{n\in\mathbb{Z}}V(F_n)\subset\mathbb{R} is not algebraic in \mathbb{A}^2(k).

Suppose S is algebraic. Then S\subset V(F) for some F\in \mathbb{R}[X]. Observe V(X)\cap S=\mathbb{Z}. Thus V(X)\cap V(F) is infinite, a contradiction.

1.11. Show that the following are algebraic sets:

(a) \{(t,t^2,t^3)\in\mathbb{A}^3(k) \ | \ t\in k\};

Observe V(X-Y^2,X-Z^3)=\{(t,t^2,t^3)\in\mathbb{A}^3(k) \ | \ t\in k\}.

(b) \{(\cos(t), \sin(t))\in\mathbb{A}^2(k) \ | \ t\in\mathbb{R}\};

It is easy to check V(X^2+Y^2-1)=\{(\cos(t), \sin(t))\in\mathbb{A}^2(k) \ | \ t\in\mathbb{R}\}.

(c) the set of points in \mathbb{A}^2(\mathbb{R}) whose polar coordinates (r,\theta) satisfy the equation r=\sin(\theta).

It is can be verified that this is the set V(X^2+(Y-1/2)^2-1/4).

1.12. Suppose C is an affine plane curve, and L is a line in \mathbb{A}^2(k), L\not\subset C. Suppose C=V(F), F\in k[X,Y] a polynomial of degree n. Show that L\cap C is a finite set of no more than n points.

Suppose L=V(aY+bX+c), where a,b,c\in k and at least one of a or b is nonzero. Without loss of generality, suppose a\ne 0. Then L=V(Y-dX-e), where d=-b/a and e=-c/a.

If (x,y)\in L then y=dx+e. Thus if (x,y)\in L\cap C then F(x,y)=F(x,dx+e)=0. Observe F(X,dX+e)\in k[X] with degree n. Thus F(X,dX+e) has at most n roots and L\cap C contains at most n points.

1.13. Show that each of the following sets is not algebraic:

(a) \{(x,y)\in\mathbb{A}^2(\mathbb{R}) \ | \ y=\sin(x)\};

Suppose A=\{(x,y)\in\mathbb{A}^2(\mathbb{R}) \ | \ y=\sin(x)\}=V(S), S\subset \mathbb{R}[X,Y]. Clearly A\ne\mathbb{A}^2(\mathbb{R}), so S\ne\emptyset. Let F\in S. Then A\subset V(F). But A\cap V(Y)=\{(2\pi t,0) \ | \ t\in\mathbb{Z}\} is infinite. Thus V(F)\cap V(Y) \supset A\cap V(Y) is infinite, a contradiction to Problem 1.12.

(b) \{(z,w)\in\mathbb{A}^2(\mathbb{C}) \ | \ |z|^2+|w|^2=1\};

Suppose A=\{(z,w)\in\mathbb{A}^2(\mathbb{C}) \ | \ |z|^2+|w|^2=1\}=V(S), S\subset\mathbb{C}(Z, W). Clearly S\ne\emptyset. Let F\in S. Observe A\cap V(W)=\{(z,0)\in\mathbb{A}^2 (\mathbb{C}) \ | \ |z|^2=1\}, an infinite set. Thus V(F)\cap V(W) is infinite, a contradiction to Problem 1.12.

(c) \{\cos(t),\sin(t),t)\in\mathbb{A}^3(\mathbb{R}) \ | \ t\in\mathbb{R}\}.

Suppose A=\{\cos(t),\sin(t),t)\in\mathbb{A}^3(\mathbb{R}) \ | \ t\in\mathbb{R}\}=V(S), S\subset \mathbb{R}[X_1,X_2,X_3]. Clearly S\ne\emptyset. Let F\in S. Note A\subset V(F). So F(\cos(t),\sin(t),t)=0 for all t\in\mathbb{R}. Thus for any \theta\in[0,2\pi),          F(\cos(\theta+2n\pi),\sin(\theta+2n\pi),\theta+2n\pi)=                 F(\cos(\theta),\sin(\theta),\theta+2n\pi). for any n\in\mathbb{Z}. But fixing \theta this gives G(X)=F(\cos(\theta),\sin(\theta),X)\in \mathbb{R}[X] with G(n)=0 for any n\in\mathbb{Z}. This is a contradiction as G must have finitely many roots in \mathbb{R}.

1.14.* Let F be a nonconstant polynomial in k[X_1,\ldots,X_n], k algebraically closed. Show that \mathbb{A}^n(k)\setminus V(F) is infinite if n\ge 1.

Suppose n=1. Then F\in k[X] and F has finitely many roots. Since k is algebraically closed, k is infinite by Problem 1.6. Thus \mathbb{A}^1(k)\setminus V(F) is infinite.

Suppose n>1. Let a_1,\ldots,a_{n-1}\in k. Then G=F(a_1,\ldots,a_{n-1},X_n)\in k[X_n]. Note G has finitely many roots b_1,\ldots,b_m. Observe          \mathbb{A}^n(k)\setminus V(F)\supset\{(a_1, \ldots,a_{n-1},x)\in\mathbb{A}^n(k) \                 | \ x\in k\setminus\{b_1,\ldots,b_m\}\}. Therefore \mathbb{A}^n(k)\setminus V(F) is infinite.

Show V(F) is infinite if n\ge 2.

Since F is nonconstant there exists i\in\{1,\ldots,n\} such that          G=F(a_1, \ldots,a_{i-1},X_i,a_{i+1}\ldots, a_n)\in k[X_i] is nonconstant, with arbitrary a_1,\ldots,a_{i-1}a_{i+1}\ldots a_n\in k. Since k is algebraically closed and G nonconstant, G has a root in k. Thus for each selection of a_1, \ldots, a_{i-1},a_{i+1},\ldots, a_n\in k, k infinite, there is a distinct root of F in \mathbb{A}^n(k). Thus V(F) is infinite.

1.15.*Let V\subset\mathbb{A}^n(k), W\subset\mathbb{A}^m(k) be algebraic sets. Show that          V\times W=\{(a_1,\ldots,a_n,b_1,\ldots,b_m) \ | \ (a_1,\ldots,a_n)\in V,                 (b_1,\ldots,b_m)\in W\} is an algebraic set.

Let V=V(A), W=V(B) with A\subset k[X_1,\ldots,X_n], B\subset k[X_1, \ldots,X_m]. Let us define \varphi:k[X_1,\ldots,X_n]\to k[X_1,\ldots,X_{n+m}] by      \varphi(F)(X_1,\ldots,X_{n})=F(X_1,\ldots,X_n). Similarly define \psi: k[X_1,\ldots,X_m]\to k[X_1,\ldots,X_{n+m}] by      \psi(F)(X_1,\ldots,X_{m})=F(X_{n+1},\ldots,X_{n+m}). It is simple to see          V\times W=V(\varphi(A) \cup \psi(B)).