# Finite schemes and secant varieties over arbitrary characteristic

@article{Buczynski2017FiniteSA, title={Finite schemes and secant varieties over arbitrary characteristic}, author={Jaroslaw Buczy'nski and Joachim Jelisiejew}, journal={Differential Geometry and Its Applications}, year={2017}, volume={55}, pages={13-67} }

## 16 Citations

### Smoothable Gorenstein Points Via Marked Schemes and Double-generic Initial Ideals

- MathematicsExp. Math.
- 2022

Abstract Over an infinite field K with we investigate smoothable Gorenstein K-points in a punctual Hilbert scheme and obtain the following results: (i) every K-point defined by local Gorenstein…

### Linear General Position (i.e. Arcs) for Zero-Dimensional Schemes Over a Finite Field

- MathematicsContemporary Mathematics
- 2021

We extend some of the usual notions of projective geometry over a finite field (arcs and caps) to the case of zero-dimensional schemes defined over a finite field Fq. In particular we prove that for…

### Smoothable zero dimensional schemes and special projections of algebraic varieties

- MathematicsMathematische Nachrichten
- 2019

We study the degrees of generators of the ideal of a projected Veronese variety v2(P3)⊂P9 to P6 depending on the center of projection. This is related to the geometry of zero dimensional schemes of…

### Elementary components of Hilbert schemes of points

- MathematicsJ. Lond. Math. Soc.
- 2019

The Bialynicki-Birula decomposition is generalized to singular schemes and an infinite family of small, elementary and generically smooth components of the Hilbert scheme of points of the affine four-space is found.

### A G ] 1 F eb 2 01 9 Elementary components of Hilbert schemes of points

- Mathematics
- 2019

Consider the Hilbert scheme of points on a higher-dimensional affine space. Its component is elementary if it parameterizes irreducible subschemes. We characterize reduced elementary components in…

### X-ranks for embedded varieties and extensions of fields

- MathematicsContributions to Mathematics
- 2022

Let X ⊂ P be a projective embedded variety defined over a field K. Results relating maximum and generic X-rank of points of P(K) and P(L) are given, where L is a field containing K. Some of these…

### Apolarity, border rank, and multigraded Hilbert scheme

- MathematicsDuke Mathematical Journal
- 2021

We introduce an elementary method to study the border rank of polynomials and tensors, analogous to the apolarity lemma. This can be used to describe the border rank of all cases uniformly, including…

### The Hilbert Scheme of 11 Points in \(\mathbb{A}^{3}\) Is Irreducible

- Mathematics
- 2017

We prove that the Hilbert scheme of 11 points on a smooth threefold is irreducible. In the course of the proof, we present several known and new techniques for producing curves on the Hilbert scheme.

### Concise tensors of minimal border rank

- MathematicsArXiv
- 2022

. We determine deﬁning equations for the set of concise tensors of minimal border rank in C m ⊗ C m ⊗ C m when m = 5 and the set of concise minimal border rank 1 ∗ -generic tensors when m = 5 , 6 .…

## References

SHOWING 1-10 OF 51 REFERENCES

### Deformations of zero-dimensional schemes and applications

- Mathematics
- 2013

In this thesis we consider the geometry of the Hilbert scheme of points in P^n, concentrating on the locus of points corresponding to the Gorenstein subschemes of P^n. New results are given, most…

### Determinantal equations for secant varieties and the Eisenbud–Koh–Stillman conjecture

- MathematicsJ. Lond. Math. Soc.
- 2013

This work puts Eisenbud's question in a more general context about the behaviour of border rank under specialisation to a linear subspace, and provides an overview of conjectures coming from signal processing and complexity theory in this context.

### Equations for secant varieties of Veronese and other varieties

- Mathematics
- 2011

New classes of modules of equations for secant varieties of Veronese varieties are defined using representation theory and geometry. Some old modules of equations (catalecticant minors) are revisited…

### ON DIFFERENCES BETWEEN THE BORDER RANK AND THE SMOOTHABLE RANK OF A POLYNOMIAL

- MathematicsGlasgow Mathematical Journal
- 2014

Abstract We consider higher secant varieties to Veronese varieties. Most points on the rth secant variety are represented by a finite scheme of length r contained in the Veronese variety – in fact,…

### Constructions of $k$-regular maps using finite local schemes

- MathematicsJournal of the European Mathematical Society
- 2019

A continuous map from R^m to R^N or from C^m to C^N is called k-regular if the images of any $k$ points are linearly independent. Given integers m and k a problem going back to Chebyshev and Borsuk…

### On the irreducibility and the singularities of the Gorenstein locus of the punctual Hilbert scheme of degree 10

- Mathematics
- 2011

### Irreducibility of the Gorenstein loci of Hilbert schemes via ray families

- Mathematics
- 2015

We analyze the Gorenstein locus of the Hilbert scheme of $d$ points on $\p n$ i.e., the open subscheme parameterizing zero--dimensional Gorenstein subschemes of $\p n$ of degree $d$ d. We give new…

### Multigraded Hilbert schemes

- Mathematics
- 2002

We introduce the multigraded Hilbert scheme, which parametrizes all homogeneous ideals with fixed Hilbert function in a polynomial ring that is graded by any abelian group. Our construction is widely…