
Mathematical Conversations
Homology Classes of Algebraic Surfaces in 4-Spaces
I will explore two questions about projections of geometric objects in 4-dimensional spaces:
(1) Let $A$ be a convex body in $\mathbb{R}^4$, and let $(p_{12}, p_{13}, p_{14}, p_{23}, p_{24}, p_{34})$ be the areas of the six coordinate projections of $A$ to $\mathbb{R}^2$. Which six numbers arise in this way?
(2) Let $S$ be an irreducible surface in $(\mathbb{P}^1)^4$, and let $(p_{12}, p_{13}, p_{14}, p_{23}, p_{24}, p_{34})$ be the degrees of the six coordinate projections from $S$ to $(\mathbb{P}^1)^2$. Which six numbers arise in this way?
The solutions to these problems are encoded in the Grassmannian $\text{Gr}(2,4)$ over the triangular hyperfield $\mathbb{T}_2$. The results suggests a general conjecture on homology classes of irreducible surfaces in smooth projective varieties. Joint with Daoji Huang, Mateusz Michalek, Botong Wang.