what is the definition of a line in $\mathbb{P}^n(k)$ + how to compute the hilbert polynomial of two intersecting lines?

Question

(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

Answer 1

Regarding the third part of your question: The intersection of two lines in $\mathbb P^3$ is generically empty (if you write down two random lines, they will be skew), but sometimes the lines will be coplanar (i.e. lie in a common plane), and then they will meet in a point (as any two lines in $\mathbb P^2$ do).

If you know that a priori that the two lines meet in a point, then they must be coplanar, and the problem is the same as for two intersecting lines in $\mathbb P^2$.

Answer 2

The line through $(a_0 : ... : a_n)$ and $(b_0 : ... : b_n)$ is uniquely parameterized by $(a_0 X + b_0 Y : a_1 X + b_1 Y : ... a_n X + b_n Y)$. Note that this precisely describes a morphism $\mathbb{P}^1 \to \mathbb{P}^n$ which is an isomorphism onto its image. I am surprised this is not given somewhere in Fulton.

Answer 3

(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.