The monic rank. (English) Zbl 1443.14062
Let \(X\) be a non-degenerate irreducible Zariski closed affine cone over an algebraically closed characteristic zero base field. In the article under review, the authors study rank questions for affine hyperplane sections of \(X\).
As one example, they define the \(k\)th open secant varieties of such affine hyperplane sections. These \(k\)th open secant varieties are used to define the \(k\)th monic secant varieties of \(X\). The authors’ main result is that the dimensions of these monic secant varieties strictly increase and then become constant.
As an illustration and application of their techniques, the authors obtain partial results that are in the direction of a conjecture of B. Shapiro. This conjecture is stated in [S. Lundqvist et al., J. Pure Appl. Algebra 223, No. 5, 2062–2079 (2019; Zbl 1409.15013)]. It predicts that every binary form of degree \(d\cdot e\) is the sum of \(d\), \(d\)th powers of degree \(e\) forms.
Finally, the authors formulate and investigate maximal monic rank questions for homogeneous varieties.
As one example, they define the \(k\)th open secant varieties of such affine hyperplane sections. These \(k\)th open secant varieties are used to define the \(k\)th monic secant varieties of \(X\). The authors’ main result is that the dimensions of these monic secant varieties strictly increase and then become constant.
As an illustration and application of their techniques, the authors obtain partial results that are in the direction of a conjecture of B. Shapiro. This conjecture is stated in [S. Lundqvist et al., J. Pure Appl. Algebra 223, No. 5, 2062–2079 (2019; Zbl 1409.15013)]. It predicts that every binary form of degree \(d\cdot e\) is the sum of \(d\), \(d\)th powers of degree \(e\) forms.
Finally, the authors formulate and investigate maximal monic rank questions for homogeneous varieties.
Reviewer: Nathan Grieve (Ottawa)
MSC:
| 14R20 | Group actions on affine varieties |
| 13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |
| 15A21 | Canonical forms, reductions, classification |
Citations:
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