I am interested in where to look to find techniques to approach the following problem:
Let $M$ be a smooth projective manifold, and let $S \subset M$ be some submanifold.
Consider on $M$ a complete linear system of divisors $|D|$ whose generic elements do not contain $S$. In this context, there is a natural constraint problem: What is the dimension of the subspace $|D|_{\supset S}$ of elements of $|D|$ that do contain $S$?
In particular, I am interested in the case where the submanifold $S$ is a curve, so that $D$ and $S$ generically intersect in a number of points.
Edit: For special arrangements, the number of resulting constraints may be fewer, and very difficult to determine in general. However I am only interested in the generic, expected result.
As a simple example, take $M=\mathbb{P}^3$, $S$ to be an intersection of two linear polynomials, and $|D|$ to be the linear system of quadratic polynomials. In this simple case, one can use something like the theory of resultants to show that $\mathrm{dim}|D|=10$ and $\mathrm{dim}|D|_{\supset S}=7$.
In more complicated situations I think that one needs, and that there should exist, a more general framework.