impanga 2026/27

The geometry of secant varieties of Segre varieties is difficult to study, partly because their definition involves taking a closure that is usually poorly understood. To approach this problem, we introduce concise secant varieties, which come equipped with projective birational morphisms to the usual secant varieties and can be viewed as partial desingularizations. A key feature of these new varieties is that their local geometry is closely related to that of well-studied moduli spaces, such as Hilbert schemes of points. We exploit this connection to construct resolutions of singularities of secant varieties in small cases. Moreover, the resulting resolutions admit natural torus actions, allowing us to apply methods of equivariant topology to investigate their finer geometry. We use these results to study the original secant varieties, including applications to the classical Salmon problem. This is joint work with Joachim Jelisiejew.

For a (smooth connected) variety $X$ over the complex numbers, the topological space $X(\mathbb{C})$ is homotopy equivalent to a finite CW-complex, and thus its fundamental group $\pi_1(X(\mathbb{C}))$ admits a finite presentation. This implies that the étale fundamental group $\pi_1^{\mathrm{\acute{e}t}} (X)$, which is a profinite group, is topologically finitely presented. In characteristic $p > 0$, the picture is much more subtle due to the abundance of Artin-Schreier covers of affine schemes. Grothendieck proved (topological) finite generation of $\pi_1^{\mathrm{\acute{e}t}} (X)$ for proper $X$ over $\bar{\mathbb{F}}_p$ and Esnault-Shusterman-Srinivas (with a small remark by L.-Srinivas-Stix to relax the assumptions) have recently shown finite presentation. We continue this study in the rigid (i.e. analytic over some non-archimedean field) and log-geometric settings. For this, we introduce the appropriate tame étale fundamental group $\pi_1^t(X/K)$ of a rigid space. The proofs rely on techniques of logarithmic geometry (extended beyond its usual scope of finitely generated monoids). This is joint work with P. Achinger, K. Hübner and J. Stix.

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.