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Anya Nordskova

Publications and source records attributed to Anya Nordskova.

9 recordsLinked to original sources
arXiv2026

A smooth projective counterexample to Bondal-Polishchuk's conjecture

For a particular smooth projective (weak Fano) threefold $X$ we show that the braid group action on the set of full exceptional collections in $D^b(X)$ is not transitive. This provides a counterexample to a conjecture of Bondal and Polishchuk from 1993. The conjecture was first disproved by Chang, Haiden, and Schroll, who constructed a family of partially wrapped Fukaya categories for which the transitivity fails. However, no counterexample of the form $D^b(X)$ for $X$ a smooth projective variety was previously known. In addition, we show that the space of Bridgeland stability conditions $Stab(X)$ on $X$ has infinitely many connected components. This is the first known example of a smooth projective variety whose space of Bridgeland stability conditions is disconnected. Finally, we apply a similar method to establish that $Stab(Y)$ for a symmetric quintic threefold $Y$ is also disconnected.

math.AG↗
arXiv2026

Groups generated by spherical twists on K3 surfaces and full exceptional collections on Fano threefolds

Let Y be a smooth K3 surface of Picard rank 1. We prove that the subgroup G of Aut D^b(Y) generated by spherical twists with respect to all spherical objects is free. Moreover, we provide a precise recipe to find free generators of G and determine the cases when G is finitely generated, depending on the degree of Y. This description in particular yields a precise classification of spherical objects in Aut D^b(Y). We apply these results to verify the first three-dimensional case of a conjecture due to Bondal and Polishchuck, namely, we establish the transitivity of the braid group action on full exceptional collections for Fano threefolds of Picard rank 1.

math.AG↗
arXiv2026

NCCRs of cones over del Pezzo surfaces

Non-commutative crepant resolutions (NCCRs) are non-commutative versions of classical crepant resolutions in algebraic geometry. For 3-dimensional terminal Gorenstein singularities Iyama and Wemyss proved that all NCCRs are connected by mutations, which may be viewed as a non-commutative analogue of Kawamata's result that all crepant resolutions are connected by flops. In this paper we prove the corresponding result for a class of canonical Gorenstein singularities which are not terminal, namely anticanonical cones over del Pezzo surfaces. More precisely, we first obtain a classification of NCCRs of anticanonical del Pezzo cones, showing that every NCCR arises from a geometric helix on the corresponding del Pezzo surface. We then prove that all such geometric helices are connected to each other by mutations, up to simple operations which include tensoring by line bundles and shifts. A crucial ingredient in our proofs is the polygons that can be associated to exceptional collections on del Pezzo surfaces following the works of Hille and Perling. We obtain some interesting observations about these polygons which may be of independent interest.

math.AG↗
arXiv2026

Derived Picard groups of symmetric representation-finite algebras of type $D$

We explicitly describe the derived Picard groups of symmetric representation-finite algebras of type $D$. In particular, we prove that these groups are generated by spherical twists along collections of $0$-spherical objects, the shift and autoequivalences which come from outer automorphisms of a particular representative of the derived equivalence class. The arguments we use are based on the fact that symmetric representation-finite algebras are tilting-connected. To apply this result we in particular develop a combinatorial-geometric model for silting mutations in type $D$, generalising the classical concepts of Brauer trees and Kauer moves. Another key ingredient in the proof is the faithfulness of the braid group action via spherical twists along $D$-configurations of $0$-spherical objects.

math.RT↗
arXiv2026

Subgroups of braid groups generated by Birman-Ko-Lee generators

We define a Young subgroup of the braid group as a subgroup generated by an arbitrary subset of the Birman-Ko-Lee generators. We give an intrinsic description of such subgroups which yields, in particular, an easy criterion to decide membership. We also give an algorithm to write an element of a Young subgroup as a product of the generators. Our methods are based on analyzing the Hurwitz action on tuples over free groups via a diagrammatic approach.

math.GR↗
arXiv2025

Local Blaschke--Kakutani ellipsoid characterization and Banach's isometric subspaces problem

We prove the following local version of Blaschke--Kakutani's characterization of ellipsoids: Let $V$ be a finite-dimensional real vector space, $B\subset V$ a convex body with 0 in its interior, and ${2\le k<\dim V}$ an integer. Suppose that the body $B$ is contained in a cylinder based on the cross-section $B \cap X$ for every $k$-plane $X$ from a connected open set of linear $k$-planes in $V$. Then in the region of $V$ swept by these $k$-planes $B$ coincides with either an ellipsoid, or a cylinder over an ellipsoid, or a cylinder over a $k$-dimensional base. For $k=2$ and $k=3$ we obtain as a corollary a local solution to Banach's isometric subspaces problem: If all cross-sections of $B$ by $k$-planes from a connected open set are linearly equivalent, then the same conclusion as above holds.

math.MG↗
arXiv2024

A note on abelian envelopes

This is a short note bridging the gap between two notions of universal abelian categories associated to exact categories, namely, Rump's quotient categories and Bodzenta-Bondal's abelian envelopes. The established connection allows us to draw several easy conclusions, in particular, we answer some (baisc) questions raised by Bodzenta and Bondal.

math.CT↗
arXiv2023

Banach's isometric subspace problem in dimension four

We prove that if all intersections of a convex body $B\subset\mathbb R^4$ with 3-dimensional linear subspaces are linearly equivalent then $B$ is a centered ellipsoid. This gives an affirmative answer to the case $n=3$ of the following question by Banach from 1932: Is a normed vector space $V$ whose $n$-dimensional linear subspaces are all isometric, for a fixed $2 \le n< \dim V$, necessarily Euclidean? The dimensions $n=3$ and $\dim V=4$ is the first case where the question was unresolved. Since the $3$-sphere is parallelizable, known global topological methods do not help in this case. Our proof employs a differential geometric approach.

math.MG↗