Giant Rat of Sumatra singularity

Question

I would be grateful for explanations of the issues raised in any of these three questions, or pointers to the relevant literature (now updated with answers):

1. How did a particular singularity come to be known as The Giant Rat of Sumatra?
   Answer: Named by Bruce & Giblin after a Sherlock Holmes reference, as explained by Michael Biro and Daniel Moskovitch.
2. What is the generic polynomial form of this singularity?
   Answer: $f_a(x,y)=x y(x-y)(x-a y)$ for a parameter $a$. This from Daniel Moskovitch's answer to the MO question, "What are some examples of colorful language in serious mathematics papers?" Here is a plot for $a=\frac{1}{4}$. Despite its formidable name, the singularity appears rather tame to the eye:
                 
3. Is there some natural equivalence relation that classifies all the giant-rat singularities into the same class (unlike Arnold's $\cal{A}$-equivalence, which in my [limited] understanding, does not).
   Answer: On p.199, Bruce & Giblin say that the polynomial above "gives uncountably many inequivalent types" (for different parameters $a$). They address my question of another equivlance relation: "topological equivalence is too weak to provide a workable theory... Instead one has to work with 'universal stratified equivalence'," a theory due to Eduard Looijenga. They go on to say, "Even for the giant rat of Sumatra one has little idea what these models are. The world is indeed not yet prepared for its story." :-)

I ask this as someone largely ignorant of singularity theory. Thanks for enlightening me (and other MO participants)!
              

Answer 1

I believe all is answered by Page 196 (and the subsequent discussion) of Curves and Singularities by J.W. Bruce and P.J. Giblin, who coined the name.

Answer 2

I'd just like to add something about the plot of the graph of the function $z=\mathrm{f}(x,y)$.

The term "singularity", in this context, does not refer to a function whose graph is singular.

The Giant Rat is a function germ $\mathrm{f} : (\mathbb{R}^2,0) \to (\mathbb{R},0)$. We're interested in the set of $(x,y) \in \mathbb{R}^2$, very close to $(0,0) \in \mathbb{R}^2$ which are sent to $0 \in \mathbb{R}$. In other words, what does $$\mathrm{f}^{-1}(0) = \{(x,y) \in \mathbb{R}^2 : \mathrm{f}(x,y)=0\}$$ look like in a small neighbourhood of the origin?

If $a \neq 0,1$ then $xy(x-y)(x-ay)=0$ are four lines all crossing at $(0,0)$. This is quite a nasty singularity. Consider, for a moment, the $D_4$ singularity $x^3-xy^2$. The zero-level set of this is three distinct real lines through the origin. Any function germ $\mathrm{g}:(\mathbb{R}^2,0) \to (\mathbb{R},0)$, whose degree three Taylor series is a cubic polynomial in $x$ and $y$ whose zero-level set is three distinct real lines through the origin is $\mathscr{R}$-equivalent to $x^3-xy^2$, i.e. there is a diffeomorphism $\mathrm{\phi} : (\mathbb{R}^2,0) \to (\mathbb{R}^2,0)$ for which $$\mathrm{g} \circ \phi = x^3 - xy^2$$ This isn't totally unexpected because any three concurrent lines can be taken to any other three concurrent lines via a linear transformation, meaning that any cubic polynomial whose zero-level set is three distinct real lines can be taken to $x^3 - xy^2$ by a linear transformation. Going back to the Giant Rat, with $a \neq 0,1$, we have four distinct real lines through the origin. Four concurrent lines can be taken to four other concurrent lines by a projective transformation if, and only if, they have the same cross ratio. The Giant Rat is so nasty because there are an uncountable number of singularity types in the family $xy(x-y)(x-ay)$ which are not $\mathscr{R}$-equivalent. The zero-level sets $xy(x-y)(x-ay)=0$ are almost all different from each other in the sense that no diffeomorphism can take $xy(x-y)(x-ay)=0$ to $xy(x-y)(x-by)=0$, where $a \neq b$. The cross ratios must be equal for there to be any hope of them being $\mathscr{R}$-equivalent. Equal cross ratios is necessary but not sufficient.