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.

Date & Time

February 12, 2025 | 6:00pm – 8:00pm

Location

Simons Hall Dilworth Room

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