Some computation I was doing reduced to counting the number of points in a algebraic subvariety of the 34-dimensional projective space over $\mathbb{F}_q$ (expressed as a polynomial in $q$). Googling a little for methods to solve this type of problems I found that this is the subject of the notorious Weil conjectures, whose proof I do not expect to understand in my lifetime.
However, my particular variety is really user-friendly, being the intersection of the zero sets of polynomials that are all of degree either 1 or 2. I would intuitively expect that in this case the counting problem is much simpler and a method of doing it has been known already around the time dr. Weil was born. Is that true and if so, can someone point me to a place in the literature where I can learn the method of solving the counting problem for these projective varieties of very low degree? Any help is appreciated.