arXiv subjects

Jakub Jagiełła

Publications and source records attributed to Jakub Jagiełła.

4 recordsLinked to original sources

arXiv2026

Geometric and homological interpretations of tensor flattenings

We relate several nonlinear flattenings of structure tensors of finite algebras and modules to their algebraic and homological invariants. Since smoothability of algebras is equivalent to their structure tensors having minimal border rank, this shows equivalence of several tensor-theoretic nonsmoothability certificates to known classical ones. Particularly, we prove two conjectures of Doležálek and Michałek relating the tangency flattening and the $3\times3$ minor flattening to dimensions of tangent spaces to the Hilbert scheme of points. In the process, we generalize the tangency flattening to a mixed regime with two tensor inputs, relating this to dimensions of $\operatorname{Ext}$ modules, and introduce the dual notion of the cotangency flattening, related to dimensions of $\operatorname{Tor}$ modules. In the case of algebras, we also realize the kernel of the dualized tangency flattening within the setting of Harrison cohomology, making the connection to the tangent space of the Hilbert scheme explicit.

math.AG↗

arXiv2026

Unrestrictions and concise secant varieties

We introduce the concise secant varieties, which are, informally speaking, modular partial desingularisations of secant varieties to Segre embeddings. More precisely, they are projective and birational to the abstract secant varieties, yet each of their points corresponds to a concise tensor of appropriate border rank (that is, to a minimal border rank tensor). We discuss implications throughout the theory of tensors, including a characterisation of border rank $\leq r$ tensors as unrestrictions of minimal border rank $r$ tensors, a characterisation of tensors with cactus rank $\leq r$, a concise versions of border apolarity including the fixed point theorem, concise Varieties of Sums of Powers, connections to defectivity and identifiability in the Segre case, to the Salmon conjecture etc.

math.AG↗

arXiv2025

Characteristic numbers of algebras

We introduce characteristic numbers of a finite commutative unital $\mathbb{C}$-algebra, which are numerical invariants arising from algebraic intersection theory. We characterize Gorenstein and local complete intersection algebras in terms of their characteristic numbers. We compute characteristic numbers for certain families of algebras. We show that characteristic numbers are constant on $\mathrm{Hilb}_d(\mathbb{A}^1)$, provide an explicit upper bound for characteristic numbers on the smoothable component of $\mathrm{Hilb}_d(\mathbb{A}^n)$ and an explicit lower bound for characteristic numbers on the Gorenstein locus of $\mathrm{Hilb}_d(\mathbb{A}^n)$ for $n \geq d-2$.

math.AG↗

arXiv2025

Classification and degenerations of small minimal border rank tensors via modules

We give a self-contained classification of $1_*$-generic minimal border rank tensors in $\mathbb{C}^m \otimes \mathbb{C}^m \otimes \mathbb{C}^m$ for $m \leq 5$. Together with previous results, this gives a classification of all minimal border rank tensors in $\mathbb{C}^m \otimes \mathbb{C}^m \otimes \mathbb{C}^m$ for $m \leq 5$: there are $107$ isomorphism classes (only $37$ up to permuting factors). We fully describe possible degenerations among the tensors. We prove that there are no $1$-degenerate minimal border rank tensors in $\mathbb{C}^m \otimes \mathbb{C}^m \otimes \mathbb{C}^m $ for $m \leq 4$.

math.AG↗