One sees that given the $SL(2,\mathbb{C})$ action on $\mathbb{C}^5$, thought of as the space of polynomials of the form,

$$a_0 x^4 + 4a_1 x^3 y + 6a_2x^2y^2 + 4a_3xy^3 + a_4 y^4$$

the ring of invariants is generated by the following functions,

$$g_2(a) = a_0a_4 - 4a_1 a_3 + 3a_2^2$$

and

$$g_3(a) = a_0a_2a_4 - a_0a_3^2 - a_1^2a_4 + 2a_1a_2a_3 - a_2^3$$

But if these $g_2$ and $g_3$ satisfy the discriminant $=0$ condition then there are inequivalent $SL(2,\mathbb{C})$ polynomials which map to the same $(g_2,g_3)$ point.

But if I look at say Theorem 5.9 in the book by Mukai then I get to see that the closure equivalent classes of orbits of the action of a linearly reductive group like $SL(2,\mathbb{C})$ on $\mathbb{C}^5$ are in bijective correspondence to the the points of $\mathbb{C}^5//SL(2,\mathbb{C})$ (which is defined as the spectrum of the invariant polynomials under $SL(2,\mathbb{C})$)

Also look at the theorem at the end of page 11 of this paper.

In the above paper "//" is defined as identifying points in the affine variety if one lies in the closure of the orbit through the other.

Are these two notions of "//" equivalent? If yes, how?

In the light of the above two theorems, can one say that the $SU(2)$ invariant polynomials among binary homogeneous quartics are in bijection with those closure equivalent classes of orbits of $SU(2)^{\mathbb{C}} = SL(2,\mathbb{C})$ which are labeled by the pairs of invariants $(g_2,g_3)$ such that $g_2^3 - 27 g_3^2 \neq 0$ ?

Hence if I am interested in only closure equivalent orbits can I just forget those pairs of values of the invariants which lie on the discriminant $0$ curve?

Conversely given a $(g_2,g_3)$ for which the above discriminant condition is satisfied can one write down the family of $SU(2)$ invariant polynomials explicitly?

I would anyway like to know how to distinguish the orbits corresponding to the discriminant $0$ condition.

In light of the various extremely helpful and references that have come up, I realize that there there is a notion of a "discriminant" for homogeneous polynomials of degree $d$ in $n$ variables. (call this space of polynomials as $P(n,d)$) This discriminant is in some sense a "homogeneous invariant" and for the $n=2$ case in which I am interested in, it generates the subalgebra of the coordinate ring over of these polynomials which is invariant under this group action (call that $\mathbb{C}[P(n=2,d=4)]^{SL(2,\mathbb{C})}$). (this is lucky!)

I guess the discriminant in this case has to be a polynomial in polynomials of homogeneous degree $4$ in $2$ variables. The above I guess implies that $\mathbb{C}[P(n=2,d=4)]^{SL(2,\mathbb{C})}$ is generated by just one such polynomial in polynomials.

I would like to know how is a ``discriminant" defined for such polynomials. (searching and asking around I am only getting definitions for the single variable case)

I want to know if knowing the generator of $\mathbb{C}[P(n=2,d=4)]^{SL(2,\mathbb{C})}$ tells me about the initial objective of knowing $P(n=2,d=4)\text{ }mod\text{ }SL(2,\mathbb{C})$

I wonder if this unique generator of the invariant subalgebra is related to the null-cone that was pointed out by Bart in his comment.

Also I would like to be pointed out if there is any mistake in what I said above!

I had recently tried asking a similar question here. But I think I could not precisely convey what I was looking for. Let me here try to give a specific situation that I need to understand coming from certain other considerations in Superconformal Quantum Field Theories.

I can think of $\mathbb{C}^5$ as being the space of all homogeneous degree $4$ polynomials in $2$ variables. On this space $SL(2,\mathbb{C})$ has the standard action.

I want to know what is the most explicit (or the best!) way to describe the quotient space thus obtained. I want to understand how do the orbits look like.

[EDIT: I was initially asking if there exists fixed subspaces etc but then from the comments I realized that I was missing the elementary fact that it is an irreducible representation! Hence nothing like this can exist.]

I tried something naive. I wrote down the most general element of $SL(2,\mathbb{C})$ using its canonical polar decomposition and then acted it on the most general homogeneous degree $4$ polynomial in $2$ variables and tried to see how the coefficients change. Unfortunately the equations are very complicated and I didn't see any hope of me being able to solve them to find the fixed points.

Apart from this specific example I would also like to know of references to simpler examples than this where a similar question is asked and answered.