# Lines in projective and affine space

I'm trying to understand lines in affine and projective space in order to solve problems 2.15 and 4.13 in Algebraic Curves by William Fulton: https://www.google.com/url?sa=t&source=web&rct=j&url=http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf&ved=2ahUKEwiI3drj-NHhAhXJxIsKHQSMC9QQFjAAegQIAhAB&usg=AOvVaw0_YBJKOU-rk2J9pX3aCJyF

Fulton defines a line trough points $$P=(a_1,...,a_n), Q=(b_1,...,b_n) \in \Bbb{A}^n$$ to be {$$(a_1+t(b_1-a_1),...,a_n+t(b_n-a_n))| t \in k$$},

and a line trough points $$P=[a_0:...:a_n], Q=[b_0,...,b_n] \in \Bbb{P}^n$$ to be {$$[\mu a_0+\lambda b_0:...:\mu a_n+\lambda b_n]| \mu, \lambda \in k$$, not both zero}.

In particular I'm trying to show (in both the affine and projective case) that a line corresponds to a linear subvariety of dimension $$m=1$$, which Fulton defines to be a variety of the form $$V=V(F_1,...,F_n)$$ ($$degF_i=1$$) that can be mapped to $$V(X_{m+1},...,X_n)$$, i.e., to $$V(X_2,...,X_n)$$ (or ($$V(X_1,...,X_n)$$ in the projective case) by an affine/projective change of coordinates.

I.e. that any line is such a linear subvariety and vice versa. Note that I'm only familiar with the formalism presented in Fulton, so a more classical approach is preferred.

• You have defined a linear subvariety, but not what you mean by a line. Can you tell us? – Mohan Apr 15 at 12:11
• Thanks, I updated my question to include these definitions. – ToricTorus Apr 15 at 12:28
• With your definition, it is easy and can you tell us what you have tried? – Mohan Apr 15 at 15:09
• Right, in the affine case it seems as though a line trough $P,Q \in \Bbb{A}^n$, is given by the vector equation $L=A+t(B-A)$, and I think I should be able to transform this to an algebrqic set by affine coorfinate changes somehow. In the projective case, It seems to me as though a line in $\Bbb{P}^n$ corresponds to a hyperplane in $\Bbb{A}^{n+1}$ trough the origin, given by the vector equation $\mu P + \lambda Q$, is this right? I guess in this case the problem boils down to expressing the parametric equation of the plane by an equation in $n+1$ variables. – ToricTorus Apr 15 at 15:59
• Hint for the affine case: $\frac{X_i - a_i}{b_i - a_i} = t = \frac{X_j - a_j}{b_j - a_j} \implies (b_j - a_j)(X_i - a_i) - (b_i - a_i)(X_j - a_j) = 0$ for all $i,j$. Setting $t = \frac{\lambda}{\mu}$ will give the projective case. – André 3000 Apr 15 at 17:18