This following is a brief summary of our research meeting on 2024-07-12.

What happened

• Aaron summarized the idea of the projective plane through the analogy with perspective lines.
• We then covered the general idea of homogeneous coordinates on ${\mathbf{P}}^2(\mathbf{R})$ and the connection with lines through the origin in ${\mathbf{R}}^3$.
• We talked about how polynomials in $x,y,z$ no longer define functions on ${\mathbf{P}}^2(\mathbf{R})$ due to the fact that the point $[a,b,c]$ really corresponds to an equivalence class of points, namely all points of the form $(\lambda a,\lambda b,\lambda c)\in {\mathbf{R}}^3$.
• We noted that if $f(x,y,z)$ is homogeneous then $f(\lambda x,\lambda y,\lambda z)={\lambda }^{d}f(x,y,z)$. We still can't evaluate $f$ at a point $[a,b,c]\in {\mathbf{P}}^2(\mathbf{R})$, but we can ask whether $f(a,b,c)=0$ for such a point. In other words, we can still talk about zero sets of homogeneous polynomials.
• We noted that a line in ${\mathbf{P}}^2(\mathbf{R})$ is a curve given by a homogeneous linear equation $Ax+By+Cz=0$.
• We talked about affine charts in ${\mathbf{P}}^2(\mathbf{R})$, which are open subsets that "look like" (i.e., are isomorphic to) ${\mathbf{R}}^2$.
• We talked about the process of homogenization, whereby we convert any polynomial $f(x,y)$ into a homogeneous polynomial $\tilde{f}(x,y,z)$; we can recover the original function $f(x,y)$ using $\tilde{f}(x,y,1)$.
• We then looked at a Exercises 4 and 5 in Section 8.1, wherein we started with an affine curve given by an equation $f(x,y)=0$, then homogenized to get a curve $\tilde{f}(x,y,z)=0$ in ${\mathbf{P}}^2(\mathbf{R})$, and then finally looked at the restriction of that projective curve to each of the three coordinate charts (by setting either $x=1$, $y=1$, or $z=1$).
• We talked a little bit about the duality between points and lines in ${\mathbf{P}}^2$, following Exercise 11 in Section 8.1.

Tasks for next meeting

• Read the first section (i.e., first three pages) of this paper. It's a senior project by former Cal Poly student Kathryn Burton, who made the first investigations into a tropical version of the duality map for projective plane conics. In her first section she reviews (without proofs!) the classical result and several different formulations/proofs.
• Investigate the classical duality map for some specific projective plane conics, to see how all of the formulations come together.