impanga 2026/27

I will discuss the geometric construction and properties of an unramified endomorphism of $\mathbb{A}^3$ that is not an isomorphism, and the construction of an exotic $\mathbb{A}^4$ whose product with $\mathbb{A}^1$ is isomorphic to $\mathbb{A}^5$. The first gives a counterexample to the Jacobian conjecture in every dimension greater than 2; the 2-dimensional case remains open. The second gives a counterexample to Zariski’s cancellation conjecture in dimension 4 and yields nonlinearizable $\mathbb{G}_m$-actions on $\mathbb{A}^n$ for $n$ greater than $4$. Both examples were discovered by AI.

Although toric varieties are Mori dream spaces and their Cox rings are as simple as possible (i.e. they are polynomial rings), the work of González-Hering-Payne-Süß shows that in general we cannot expect that a projectivized vector bundle over a toric variety retains the Mori dream property. On the other hand, earlier work of Hausen-Süß gives an explicit presentation of the Cox ring for the projectivized tangent bundle over a smooth toric variety, which is finitely generated (so in this case we retain the Mori dream property). This dichotomy brings natural questions: when does the projectivized tangent bundle over a smooth Mori dream space remains a Mori dream space? What is its Cox ring? I will report on a work in progress (joint with Jeong-Seop Kim, Sunchon National University) where we provide an answer for del Pezzo surfaces along with some connections to well-known problems in geometry of Fano varieties.