Fulton 2.3

2.14.* A set $V\subset\mathbb{A}^n(k)$ called a linear subvariety of $\mathbb{A}^n(k)$ if $V=V(F_1,\ldots,F_r)$ for some polynomials $F_i$ of degree $1.$

(a) Show that if $T$ is an affine change of coordinates on $\mathbb{A}^n$ the $V^T$ is also a linear subvariety of $\mathbb{A}^n.$

Let $T=(T_1,\ldots,T_n)$ be an affine change of coordinates, and $V=V(F_1,\ldots,F_r)$ a linear subvariety of $\mathbb{A}^n.$ Then $I(V)=(F_1,\ldots,F_r),$ since $F_1,\ldots,F_r$ are of degree $1.$ Note that $F_1^T,\ldots,F_r^T$ are also degree $1.$ So $I(V)^T=(F_1^T,\ldots,F_r^T).$ therefore $V^T=V(I(V)^T)=V(F_1^T,\ldots,F_r^T).$

(b) If $V\ne\emptyset,$ show that there is an affine change of coordinates $T$ of $\mathbb{A}^n$ such that $V^T=V(X_{m+1}\ldots,X_n).$ So $V$ is a variety.

Suppose $r=1,$ that is, $V=V(F).$ Let $F=\sum_{i=1}^na_iX_i+a_0.$ Let $T&39;$ be an affine change of coordinates relabeling varables such that $F^U=\sum_{i=d+1}^na_iX_i+a_0$ with $a_{d+1},\ldots,a_n\in k\setminus\{0\}.$ By Lemma 2.8 there exists an affine change of coordinates $U$ on $\mathbb{A}^{n-d},$ defined by polynomials $U_d,\ldots,U_n\in k[X_d,\ldots,X_n]$ such that $(F^{T&39;})^{U}=X_n.$ Let $T&39;&39;$ be the affine change of coordinates defined by $T_i=X_i$ if $i<d$ and $T_i=U_i$ if $i\ge d.$ Let $I_d$ be the $d\times d$ identity matrix and observe that $T&39;&39;$ is represented by the following upper triangular matrix $T&39;&39; = \begin{bmatrix} I_d & 0\\ 0 & U \end{bmatrix}$ Thus $T&39;&39;$ is invertible and $T=T&39;&39;\circ T&39;$ is an affine change of coordinates such that $F^{T}=X_n.$ Thus $V^{T}=V(X_n).$

Let $r\ge 1.$ Suppose for any $F_1,\ldots,F_r\in k[X_1,\ldots,X_n]$ there exists affine change of coordinates $T$ such that $V(F_1,\ldots,F_r)^T=V(X_{m+1},\ldots,X_n)$ for some $m\ge 0.$ Let $F_1,\ldots,F_{r+1}\in k[X_1,\ldots,X_n]$ and $V=V(F_1,\ldots,F_{r+1}).$ By the inductive hypothesis there exists affine change of coordinates $T&39;$ such that $V(F_1,\ldots,F_{r})^T=V(X_{m+1},\ldots,X_n).$ Therefore $V^{T&39;}=V(F_{r+1}^{T&39;},X_{m+1},\ldots,X_n).$ If $F_{r+1}^{T&39;}\in (X_{m+1},\ldots,X_n)$ we are done. Otherwise, let us select $G\in k[X_1,\ldots,X_{m}]$ such that $F_{r+1}^{T&39;}+(X_{m+1},\ldots,X_n)=G+(X_{m+1},\ldots,X_n).$ Then $V^{T&39;}=V(G,X_{m+1},\ldots,X_n).$ By the same method as the base case, we may construct an affine change of coordinates $T&39;&39;$ on $\mathbb{A}^n,$ such that $G^{T&39;&39;}=X_{m}$ and $X_i^{T&39;&39;}=X_i$ for all $i>m.$ Thus if we define $T=T&39;&39;\circ T&39;,$ then $V^T=V(X_{m},\ldots,X_n).$

(c) Show that the $m$ appearing in part (b) is independent of the choice of $T.$ Thus $V$ is isomorphic to $\mathbb{A}^m(k).$

Suppose there exist change of coordinates $T,U$ such that $V^T=V(X_{m+1},\ldots,X_n)$ and $V^U=V(X_{d+1},\ldots,X_n).$ Observe that $T|_{V^T}:V^T\to V$ and $U|_{V^U}:V^U\to V$ are each isomorphisms. So $k[X_1,\ldots,X_m]\cong\Gamma(V^T)\cong\Gamma(V)\cong\Gamma(V^U)\cong k[X_1,\ldots,X_d].$ Therefore $m=d.$ Moreover, $\Gamma(\mathbb{A}^m)\cong k[X_1,\ldots,X_m]\cong\Gamma(V).$

2.15.* Let $P=(a_1,\ldots,a_n),$ $Q=(b_1,\ldots,b_n)$ be distinct points in $\mathbb{A}^n.$ The line through $P$ and $Q$ is defined to be $L=\{ (a_1+t(b_1-a_1),\ldots,a_n+t(b_n-a_n)) \ | \ t\in k\}.$

(a) Show that if $L$ is the line through $P,Q$ and $T$ is an affine change of coordinates, then $T(L)$ is the line through $T(P),T(Q).$

Let $T_i=\sum_{j=1}^nc_{i,j}X_i+c_{i,0}.$ Note that $T_i(P)=\sum_{j=1}^n c_{i,j}a_j+c_{i,0}$ and $T_i(Q)=\sum_{j=1}^nc_{i,j}b_j+c_{i,0}.$ Let $M$ be the line between $T(P)$ and $T(Q).$ Let $t\in k.$ Let $O=\left(\sum_{j=1}^nc_{1,j}(a_j+t(b_j-a_j)+c_{1,0},\ldots, \sum_{j=1}^nc_{n,j}(a_j+t(b_j-a_j)+c_{n,0}\right)\in M$ Let $O&39;=(a_1+t(b_1-a_1),\ldots,a_n+t(b_n-a_n))\in L.$ Observe $T(O&39;)=O$ and thus $T$ is a bijection between $L$ and $T(L)=M.$

(b) Show that a line is a linear subvariety of dimension 1, and that a linear subvariety of dimesnion 1 is the line through any two of its points.

Let $L$ be the line between $P$ and $Q$ defined above. Observe that $(X_1,\ldots,X_n)\in L$ if and only if there exists $t\in k$ such that $X_i=a_i+t(b_i-a_i)$ every $i.$ Since $P,Q$ are distinct points, there must exist $i\in\{1,\ldots,n\}$ such that $a_i\ne b_i.$ Therefore we may solve for $t$ and find $t=\frac{X_i}{b_i-a_i}- \frac{a_i}{b_i-a_i}.$ So $(X_1,\ldots,X_n)\in L$ if and only if $F_j=(b_i-a_i)(X_j-a_j)-(b_j-a_j)(X_i-a_i)=0$ for all $j\ne i.$ Therefore $L=V(\{F_j \ | \ j\in\{1,\ldots,n\}\setminus\{i\}\}).$ Thus $L$ is a linear subvariety of $\mathbb{A}^n.$ Observe that $\{F_j \ | \ j\in\{1,\ldots,n\}\setminus\{i\}\}$ is a minimal generating set (Definition 2.10) for $I(L).$ Therefore, by Lemma 2.11, $L$ has dimension $1.$

Suppose $V$ is a linear subvariety of dimension $1.$ Then there exists an affine change of coordinates $T$ such that $V^T=V(X_2,\ldots,X_n).$ Note that $V^T$ is the line between $P=(0,0,\ldots,0)$ and $Q=(1,0,\ldots,0).$ Thus, by part (a), $T(V^T)=V$ is the line between $T(P)$ and $T(Q).$

By Lemma 2.11, in $\mathbb{A}^2$ a variety has dimension 1 if and only if it is a hyperplane. Thus, by part (b), a line is the same thing as a hyperplane.

(d) Let $P,P&39;\in\mathbb{A}^2,$ $L_1,L_2$ two distinct lines through $P,$ $L_1&39;,L_2&39;$ distinct lines through $P&39;.$ Show that there is an affine change of coordinates $T$ of $\mathbb{A}^2$ such that $T(P)=P&39;$ and $T(L_i)=L_i&39;,$ $i=1,2.$

Let $U_P,U_{P&39;}$ be the translations sending $P,P&39;$ to the origin. Let $v_1,v_2$ be unit vectors along $U_P(L_1),U_P(L_2).$ Similarly let $u_1,u_2$ be unit vectors along $U_{P&39;}(L_1&39;),U_{P&39;}(L_2&39;).$ Note that $\{v_1,v_2\}$ and $\{u_1,u_2\}$ are each a basis for $\mathbb{A}^2.$ Let $T&39;$ be the linear map defined by $v_i\mapsto u_i.$ Then $T=U_{P&39;}^{-1}\circ T&39;\circ U_P$ is an affine change of coordinates such that $T(P)=P&39;$ and $T(L_i)=L_i&39;,$ $i=1,2.$

2.16. Give $\mathbb{A}^n(\mathbb{C})=\mathbb{C}^n$ the usual topology.

(a) Show that $\mathbb{C}^n\setminus S$ is path-connected for any finite set $S.$

Let $P,Q\in\mathbb{C}^n\setminus S$ be distinct points. Let $R$ be the set of all points on lines between $P$ or $Q$ and a point in $S,$ lines in $\mathbb{R}^{2n},$ not dimension 1 linear subvarieties of $\mathbb{A}^n(\mathbb{C}).$ Since $S$ is finite, $R\ne\mathbb{C}^n.$ Therefore we may pick a point $O\in\mathbb{C}^n\setminus R.$ Observe that neither the line segment from $P$ to $O$ nor the line segment from $O$ to $Q$ contain a point in $S.$ Thus there is a path from $P$ to $Q$ in $\mathbb{C}^n\setminus S.$

(b) Let $V$ be an algebraic set in $\mathbb{A}^n(\mathbb{C}).$ Show that $\mathbb{A}^n(\mathbb{C})\setminus V$ is path connected.

If $V=\mathbb{A}^n(\mathbb{C})$ then $\mathbb{A}^n(\mathbb{C})\setminus V=\emptyset$ and is therefore path connected. Suppose $V\subsetneq\mathbb{A}^n(\mathbb{C}).$ Let $P,Q\in\mathbb{A}^n(\mathbb{C})\setminus V,$ $P\ne Q,$ and let $L$ be the line between $P$ and $Q,$ in the sense of linear subvariety of dimension 1. By Lemma 2.12, $V\cap L$ is finite. Moreover, $L$ is isomorphic to $\mathbb{A}^1(\mathbb{C}).$ Note that polynomial maps are continuous with respect to the topology on $\mathbb{C}^n,$ so $L$ is in fact homeomorphic to $\mathbb{A}^1(C).$ Therefore $L\setminus V$ is homeomorphic to $\mathbb{C}$ with a finite set removed. Thus $L\setminus V$ is path connected by part (a). Therefore $\mathbb{A}^n(\mathbb{C})\setminus V$ is path connected.