Let $\mathcal{P}\_d\cong\mathbb{A}^d$ denote the set of monic degree $d$ polynomials defined over an algebraically closed field of characteristic $0$, where we identify $f(x)$ with its coefficients. The multiplicities of the roots of $f(x)\in\mathcal{P}\_d$ defines a partition $\pi(f)$ of $d$. For example, if $f(x)=(x-\alpha)^d$, then $\pi(f)=(d)$, and if $f(x)$ has distinct roots, then $\pi(f)=(1^d)$.


For any partition $\sigma$ of $d$, the set
$$ \mathcal{P}\_d(\sigma) := \bigl\{ f\in\mathcal{P}\_d : \pi(f)=\sigma\bigr\} $$
is a quasiprojective subvariety of $\mathbb{A}^d$. (This follows from elimination theory.) For example, $\mathcal{P}\_d(d)$ is a curve, while $\mathcal{P}\_d(1^{d-2},2)$ is an open subset of the discriminant locus $\bigl\{f\in\mathcal{P}\_d:\operatorname{Disc}(f)=0\bigr\}$.


* Do these varieties $\mathcal{P}\_d(\sigma)$ have a name? My best guess was *discriminantal variety*, but that term does not seem to be in use.
* Where have these varieties been studied? Specific references would be appreciated.



The locus of polynomials with specified root multiplicities
I may be wrong but I think "coincident root loci" mentioned in Gjergjji's comment was coined by my coauthor Jaydeep Chipalkatti. The ideals of such varieties are in general poorly understood. 


For a general study see:


1. J. Chipalkatti, ["On equations defining Coincident Root loci"](https://www.sciencedirect.com/science/article/pii/S0021869303003363), J. Algebra **267** (2003), no. 1, 246-271.
2. J. Chipalkatti, ["Invariant equations defining coincident root loci"](https://link.springer.com/article/10.1007/s00013-004-1191-z), Archiv der Math. **83** (2004), no. 5, 422-428.
3. H. Lee and B. Sturmfels, ["Duality of multiple root loci"](https://www.sciencedirect.com/science/article/pii/S0021869315004822), J. Algebra **446** (2016), 499-526.


For particular cases, the following might be also of interest.


1. A. Abdesselam and J. Chipalkatti, ["Brill–Gordan loci, transvectants and an analogue of the Foulkes conjecture"](https://www.sciencedirect.com/science/article/pii/S0001870806000909), Adv. Math. **208** (2007), no. 2, 491-520. It has a description of the ideal for a partition with two equal parts.
2. A. Abdesselam and J. Chipalkatti, ["The bipartite Brill-Gordan locus and angular momentum"](https://link.springer.com/article/10.1007/s00031-005-1111-8), Transformation Groups **11** (2006),
no. 3, 341-370. It has a description of the ideal for a partition with two unequal parts.
3. A. Abdesselam and J. Chipalkatti, ["On Hilbert Covariants"](https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/on-hilbert-covariants/080B5B7777CFA44DA20622B1598A5C0A)
Canadian J. Math. **66** (2014), no. 1, 3-30. It has several set-theoretic systems of equations for the varieties corresponding to rectangular partitions, and a conjecture about minimal degree of generators for the ideals. This is an $SL\_2$ analogue/toy version of the Foulkes-Howe conjecture.


There is also work about these varieties from a topological point of view, e.g.,
F. Napolitano,
["On some topological invariants of algebraic functions associated to the Young stratification of polynomials"](https://www.sciencedirect.com/science/article/pii/S016686410300107X)
Topology Appl. **134** (2003), no. 3, 189-201. 


The relation to Hurwitz stacks is studied in: J. Bertin and M. Romagny,
["Champs de Hurwitz"](http://www.numdam.org/item/MSMF_2011_2_125-126__1_0/), Mémoires SMF, no. 125-126 (2011), 219 p. See also [here](https://arxiv.org/abs/math/0701680) for the arXiv version. The relevant result is Theorem 9.16 relating coincident root loci and Hurwitz stacks of cyclic coverings of $\mathbb{P}^1$.


Also, an interesting geometric approach to these varieties is in: G. Katz ["How tangents solve algebraic equations, or a remarkable geometry of discriminant varieties"](https://www.sciencedirect.com/science/article/pii/S0723086903800026), Expositiones Math. **21** (2003),
no. 3, 219-261.


Of course, this is by no means an exhaustive bibliography. There are many references I left out in this short MO post. 


