Back to the circles

We now go back to our problem: collapsing all the three circles $C_1$, $C_2$ and $C_3$ to a point, we jumped from 8 tangent circles to the whole plane of possibilities.

$\includegraphics[scale=0.2]{Images/all8circles/all8circles-0}$

Remark 5.1

In fact we have two infinite planes of possibilities! Indeed, we can see this by thickening the collapse point, the tangent circles can appear either from the left or from the right

$\includegraphics[scale=0.4]{Images/doubleplane.png}$

So not only we jumped from 8 to infinitely many tangent planes, we actually have two infinity planes. How to can we retract the 8 circles from the first example?

Solution 5.2

The solution is to take the Apollonius-Toën derived intersection

\[ \mathbf{X}=Z_{C_1}\cap ^{\mathbf{dSch}} Z_{C_2}\cap ^{\mathbf{dSch}} Z_{C_3}\hspace{1cm} C_1=C_2=C_3=\ast \]

inside the projective space $\mathbb {P}^3$. The result is a derived (double) projective plane - or more precisely - a derived scheme whose underlying classical scheme is a double projective plane $\mathbb {P}^2$. We saw in Serre’s formula that redundancies have an interpretation in terms of generators of homological degree 1. In the current case, and since we are self-intersecting twice, we need to add generators $\epsilon _1$ and $\epsilon _2$ accounting for each repetition. Moreover, since we have a multiplicative structure, the product $\epsilon _1.\epsilon _2$ gives a new element in homological degree 2.

$\includegraphics[scale=0.35]{Images/brute}$

The new extra derived structure subtracts the double infinity and retraces the 8 circles algebraically. In order to see this happening we need to compute the derived fundamental class (Kontsevich 95, Fontanine-Kapranov 2009, Khan 2019) which is given by the Chern character of the derived structure sheaf:

\[ \underbrace{[\mathbf{X}]}_{\textbf{Derived Fundamental Class}}\hspace{-1cm}:=\mathrm{Ch}[2\mathcal{O}-4\mathcal{O}(-2)+2\mathcal{O}(-4)]\cap [\mathbb {P}^2]=8.[pt] \in H_0(\mathbb {P}^2) \]

In summary, the solution to the Apollonius-Toën problem is given by

\[ \mathbf{X}=Z_{C}\cap ^{\mathbf{dSch}} Z_{C}\cap ^{\mathbf{dSch}} Z_{C} \]

which is a derived projective plane and whose derived structure corrects the counting:

it is of virtual dimension zero, which means it behaves like a point
its fundamental class indicates the point has multiplicity 8.