TITLE: what is the definition of a line in $\mathbb{P}^n(k)$ + how to compute the hilbert polynomial of two intersecting lines?
QUESTION [5 upvotes]: (1) I have never studied any projective/affine geometry or algebraic curves. I'd like to see a clear definition of a line in the projective space $\mathbb{P}^n(k)$, since I need it for my algebraic geometry study.
(2) I'm guessing a line is the variety $\mathcal{V}(f_1,\ldots,f_{n-1})$, where $f_i$ are linear homogenous polynomials, whose coefficients form a $n\times n\!-\!1$ matrix of full rank. Yes or no?
Another guess would be, that a line in $\mathbb{P}^n(k)$ is uniquely determined by two points $a\!=\![a_0\!:\!\ldots\!:\!a_n]$, $b\!=\![b_0\!:\!\ldots\!:\!b_n]$, such that the matrix $\begin{bmatrix} a \\ b \end{bmatrix}$ is of rank $2$. But how is such a line parametrized? Is any of my two attempts of a definition correct?
(3) What are the defining equations of two intersecting lines in $\mathbb{P}^3$? And now, most importantly: how can I compute the Hilbert polynomial of such a variety?
For such an elementary concept, one would expect it to be the first object defined, but to my annoyance and frustration, I have yet to see an official definition. I have the book Introduction to Algebraic Geometry (Hassett), as well as Algebraic Curves (Fulton) as my main source. Any references would be highly desirable.
thank you

REPLY [2 votes]: (2) Your guess is correct, except I think it is an $(n-1)\times (n+1)$ matrix. Think of your line as the intersection of $n-1$ hyperplanes in non-degenerate position.
(3) Assuming you meant the union of two lines $L_1, L_2$, then the defining variety is given by $I_1I_2$.
As for the Hilbert poly., use:
$$0 \to \mathcal O_{L_1\cup L_2} \to \mathcal O_{L_1}\oplus \mathcal O_{L_2} \to \mathcal O_{L_1\cap L_2} \to 0$$ 
and the fact that Hilbert poly. are additive. 
Details added: the two terms on the right of the above sequence can be calculated easily. Think of a line as $Proj(k[x,y])$ and a point (since you know $L_1,L_2$ intersect at a point) as $Proj(k[x])$. I am sure you can find the Hilbert polynomials of those rings.