Inside the complex projective plane $\mathbb{CP}^2$, a linear equation defines a complex projective line $\mathbb{CP}^1$. But a quadratic equation also defines a $\mathbb{CP}^1$ $\big($recalling that $\mathbb{CP}^1$ is topologically a sphere, this is clear from the degree genus relation $g = \frac{(d-1)(d-2)}{2}$$\big)$.
However, if we want to use algebraic geometry to describe objects on the two hypersurfaces, the two situations are not equivalent. In the first case, taking the intersection with a second linear equation gives a point, but in the second case this gives two points. More generally, we cannot pick out an odd number of points on the $\mathbb{CP}^1$ with polynomials on the ambient space. Hence, from the perspective of the ambient space we are 'missing' half the divisors on the hypersurface.
Clearly this is a simple example of a more general phenomenon, in which the lattice of divisors of a complete intersection does not descend from that of the ambient space, despite the lattice ranks being equal. My question is whether there is still in these cases some way to make use of the tools of algebraic geometry to describe the 'missing' divisors.
A different but perhaps motivating situation is when the effective divisors on a complete intersection do not descend from those of the ambient space, but it is still possible to describe them, by using ineffective divisors on the ambient space which become effective when restricted to the complete intersection.