# REU Meeting - 2024-07-12

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**onand the connection with lines through the origin in . - We talked about how polynomials in
no longer define functions on due to the fact that the point really corresponds to an equivalence class of points, namely all points of the form . - We noted that if
is **homogeneous**then. We still can't evaluate at a point , but we *can*ask whetherfor such a point. In other words, we can still talk about zero sets of homogeneous polynomials. - We noted that a
**line**inis a curve given by a homogeneous linear equation . - We talked about
**affine charts**in, which are open subsets that "look like" (i.e., are isomorphic to) . - We talked about the process of
**homogenization**, whereby we convert any polynomialinto a homogeneous polynomial ; we can recover the original function using . - We then looked at a Exercises 4 and 5 in Section 8.1, wherein we started with an affine curve given by an equation
, then homogenized to get a curve in , and then finally looked at the restriction of that projective curve to each of the three coordinate charts (by setting either , , or ). - We talked a little bit about the duality between points and lines in
, 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.