This seems like a very basic question to me and I am certain that people studied it a lot. It is for sure related to deformation theory and families of varieties, but I am not sure how these fields would answer it (and I know very little of them).
Say, we are given polynomials $f_{1,(a_1,\ldots,a_n)}(x_1,\ldots,x_d),\ldots,f_{m,(a_1,\ldots,a_n)}(x_1,\ldots,x_d)$ (can be assumed to be homogeneous) over a field $k$ (algebraically closed, if you want), in the variables $x_i$ and with parameters $a_j \in k$. So, changing the parameters changes the polynomials and therefore the variety $V_{\underline a}$ they define.
Q1: Is it possible to describe the set $A$ of all points $\underline a$ such that
- $V_{\underline a}$ is smooth,
- $V_{\underline a}$ is irreducible,
- $V_{\underline a}$ is contained in some given variety $W$,
- an irreducible component of $V_{\underline a}$ is contained in $W$,
- $V_{\underline a}$ contains a given point,
- $V_{\underline a}$ satisfies some other interesting property?
Q2: Is such a set $A$ usually a variety itself? If yes, how does one extract its equations from the equations of $V$?
It would be very helpful for me if you could propose to me a good reference for such questions, especially with a view into the direction of applications to explicit families of varieties, e.g., defined by two quadratic equations, each in four parameters, where the parameter sets are disjoint, etc.
Thank you in advance!