We are asked to compute the Hilbert function of two non-intersecting lines in $\mathbb{P}^3$. I'm not sure where to start here. Any hints or even a sketch of a full solution would be greatly appreciated.
In general, how do we go about finding the Hilbert function of lines in the projective space of arbitrary dimensions? I was thinking about finding an ideal that generates the lines and then computing the function normally by counting the number of monomials in the coordinate ring but not in the ideal. What is the most efficient way to do this? Let me know if you need any more context.
Take the two lines to be $ x=y=0 $ and $ z=w=0 $ WLOG. (The general case is exactly the same) Then the ideal of the two lines is $ J = (xz,xw,yz,yw) $ and now let's count monomials of degree $ d $ in $ \mathbb{C}[x,y,z,w]/J $ as you suggested. The only surviving monomials are $ x^d, y^d, z^d, w^d $ and $ x^ay^b, z^aw^b $ with $ a+b = d, a,b > 0 $. This is a count of $ 4 + 2(d-1) = 2d+2 $. So the Hilbert polynomial is $ p(t) = 2(t + 1) $ which is the right answer because it's twice the Hilbert polynomial of a line.