ag.algebraic geometry - How does one find vanishing algebraic cycles?

I have a question, related to what I asked before.
Let's consider a smooth hyperplane section $X$ of a smooth projective variety $Y$ over $mathbb C$.
According to Weak Lefschetz theorem, cohomology groups of $X$ coincide with those of $Y$ in all dimensions except for the middle one. In the middle dimensions the pull-back $i^*: H^d(Y) to H^d(X)$ is injective, but not surjective, and the "new" cycles on $X$ are called vanishing cycles. (The reason for such a name is that these "new" cycles will vanish when we approach singular fibers on the Lefschetz pencil.)
Vanishing cycles also can be described as the ones that live in the kernel of $i_*: H^d(X) to H^{d+2}(Y)$.

Let's consider the case when $X$ is even-dimensional, so that we can hope that the vanishing cycles are algebraic.

For example, for a smooth even-dimensional quadric in $mathbb {CP}^n$ there exist one vanishing cycle - it is a difference $[E_1] - [E_2]$ of two maximal linear subspaces from different classes.

Another example I thought about is a smooth cubic surface in $mathbb {CP}^3$, vanishing cycles here are generated by differences of pairs of lines $[l_1] - [l_2]$ lying on the cubic.

Now I wanted to ask, what are other examples people have in mind?
I'm interested in the case, when vanishing cycles actually are algebraic.

Is there a general method to describe such cycles in concrete situations (like MG(3,6))?

Thanks

EDIT: Probably winter break is not a best time to start a bounty...