Algebraic curves, rich points, and doubly-ruled surfaces
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2015
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866917596124676096 |
|---|---|
| author | Guth, Larry Zahl, Joshua |
| author_facet | Guth, Larry Zahl, Joshua |
| contents | We study the structure of collections of algebraic curves in three dimensions that have many curve-curve incidences. In particular, let $k$ be a field and let $\mathcal{L}$ be a collection of $n$ space curves in $k^3$, with $n<\!\!<(\operatorname{char}(k))^2$ or $\operatorname{char}(k)=0$. Then either A) there are at most $O(n^{3/2})$ points in $k^3$ hit by at least two curves, or B) at least $Ω(n^{1/2})$ curves from $\mathcal{L}$ must lie on a bounded-degree surface, and many of the curves must form two "rulings" of this surface.
We also develop several new tools including a generalization of the classical flecnode polynomial of Salmon and new algebraic techniques for dealing with this generalized flecnode polynomial. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1503_02173 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | Algebraic curves, rich points, and doubly-ruled surfaces Guth, Larry Zahl, Joshua Algebraic Geometry Computational Geometry Combinatorics We study the structure of collections of algebraic curves in three dimensions that have many curve-curve incidences. In particular, let $k$ be a field and let $\mathcal{L}$ be a collection of $n$ space curves in $k^3$, with $n<\!\!<(\operatorname{char}(k))^2$ or $\operatorname{char}(k)=0$. Then either A) there are at most $O(n^{3/2})$ points in $k^3$ hit by at least two curves, or B) at least $Ω(n^{1/2})$ curves from $\mathcal{L}$ must lie on a bounded-degree surface, and many of the curves must form two "rulings" of this surface. We also develop several new tools including a generalization of the classical flecnode polynomial of Salmon and new algebraic techniques for dealing with this generalized flecnode polynomial. |
| title | Algebraic curves, rich points, and doubly-ruled surfaces |
| topic | Algebraic Geometry Computational Geometry Combinatorics |
| url | https://arxiv.org/abs/1503.02173 |