# Problem from Algebraic Curves by Fulton

Suppose $V$ is a variety in $\mathbb{P}^n$ and $V \supset H_\infty$. Show that $V= \mathbb{P}^n$ or $V = H_\infty$. If $V= \mathbb{P}^n, V_* = \mathbb {A}^n$ while if $V = H_\infty, V_* = \emptyset$.

My attempt: $H_\infty = \mathbb{P}^n$\ $U_{n+1} \subset V \subset \mathbb{P}^n$. Suppose $v \in V$. I am trying to prove that $v \notin U_{n+1}$.

Any help is much appreciated. Thanks in advance for any replies.

Notation: $\mathbb{P}^n =$ projective $n$ space. $H_\infty =$ Hyperplace at infinity. $V_* = V(I^*)$ where $I^* = \{ (F_*)|F \in I \}$. $F_* = F(X_1,...,X_n,1)$. $U_{n+1} = \{ [x_1:...:x_{n+1}] \in \mathbb{P}^n | x_{n+1} \neq 0 \}.$

• Please explain your notations. – user379195 May 9 '17 at 4:23

1. Helpful fact: If $X$ and $Y$ are varieties (in particular, irreducible algebraic sets cut out by a radical ideal) of the same dimension, then $X \subset Y$ implies that $X = Y$.
2. There are two options, $dim V = n$ or $dim V = n -1$. In the first case, 1. implies that $P^n = V$. In the second case, $1.$ implies that $H_{\infty} = V$.