The strength for line bundles

@article{Ballico2021TheSF,
  title={The strength for line bundles},
  author={Edoardo Ballico and Emanuele Ventura},
  journal={MATHEMATICA SCANDINAVICA},
  year={2021}
}
We introduce the strength for sections of a line bundle on an algebraic variety. This generalizes the strength of homogeneous polynomials that has been recently introduced to resolve Stillman's conjecture, an important problem in commutative algebra. We establish the first properties of this notion and give some tool to obtain upper bounds on the strength in this framework. Moreover, we show some results on the usual strength such as the reducibility of the set of strength two homogeneous… 
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References

SHOWING 1-10 OF 22 REFERENCES
On the strength of general polynomials
A slice decomposition is an expression of a homogeneous polynomial as a sum of forms with a linear factor. A strength decomposition is an expression of a homogeneous polynomial as a sum of reducible…
Polynomials and tensors of bounded strength
Notions of rank abound in the literature on tensor decomposition. We prove that strength, recently introduced for homogeneous polynomials by Ananyan–Hochster in their proof of Stillman’s conjecture…
Topological noetherianity for cubic polynomials
Let P3(k∞) be the space of cubic polynomials in infinitely many variables over the algebraically closed field k (of characteristic ≠ 2, 3). We show that this space is GL∞-noetherian, meaning that any…
Properties of High Rank Subvarieties of Affine Spaces
We use tools of additive combinatorics for the study of subvarieties defined by {\it high rank} families of polynomials in high dimensional $\mathbb{F} _q$-vector spaces. In the first, analytic part…
GENERALIZED DIVISORS AND REFLEXIVE SHEAVES
We introduce the notions of depth and of when a local ring (or module over a local ring) is “S2”. These notions are found in most books on commutative algebra, see for example [Mat89, Section 16] or…
Moduli of curves
Parameter spaces: constructions and examples * Basic facts about moduli spaces of curves * Techniques * Construction of M_g * Limit Linear Series and the Brill-Noether Theory * Geometry of moduli…
On Ranks of Polynomials
Let V be a vector space over a field k, P : V → k, d ≥ 3. We show the existence of a function C(r, d) such that rank(P) ≤ C(r, d) for any field k, char(k) > d, a finite-dimensional k-vector space V…
Algebraic Geometry
Introduction to Algebraic Geometry.By Serge Lang. Pp. xi + 260. (Addison–Wesley: Reading, Massachusetts, 1972.)
Big polynomial rings and Stillman’s conjecture
Ananyan-Hochster's recent proof of Stillman's conjecture reveals a key principle: if $f_1, \dots, f_r$ are elements of a polynomial ring such that no linear combination has small strength then $f_1,…
Subvarieties of general hypersurfaces in projective space
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