Classification of secant defective manifolds near the extremal case
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2011
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866910425027706880 |
|---|---|
| author | Han, Kangjin |
| author_facet | Han, Kangjin |
| contents | Let $X\subset ¶^N$ be a nondegenerate irreducible closed subvariety of dimension $n$ over the field of complex numbers and let $SX\subset¶^N$ be its secant variety. $X\subset¶^N$ is called `secant defective' if $\dim(SX)$ is strictly less than the expected dimension $2n+1$. In \cite{Z1}, F.L. Zak showed that for a secant defective manifold necessarily $N\le{n+2 \choose n}-1$ and that the Veronese variety $v_2(¶^n)$ is the only boundary case. Recently R. Mu$\tilde{\textrm{n}}$oz, J. C. Sierra, and L. E. Solá Conde classified secant defective varieties next to this extremal case in \cite{MSS}.
In this paper, we will consider secant defective manifolds $X\subset¶^N$ of dimension $n$ with $N={n+2 \choose n}-1-ε$ for $ε\ge0$. First, we will prove that $X$ is a $LQEL$-manifold of type $δ=1$ for $ε\le n-2$ (see Theorem \ref{main_thm}) by showing that the tangential behavior of $X$ is good enough to apply Scorza lemma. Then we will completely describe the above manifolds by using the classification of conic-connected manifolds given in \cite{IR1}. Our method generalizes previous results in \cite{Z1,MSS}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1108_4264 |
| institution | arXiv |
| publishDate | 2011 |
| record_format | arxiv |
| spellingShingle | Classification of secant defective manifolds near the extremal case Han, Kangjin Algebraic Geometry 14MXX, 14NXX, 14M22 Let $X\subset ¶^N$ be a nondegenerate irreducible closed subvariety of dimension $n$ over the field of complex numbers and let $SX\subset¶^N$ be its secant variety. $X\subset¶^N$ is called `secant defective' if $\dim(SX)$ is strictly less than the expected dimension $2n+1$. In \cite{Z1}, F.L. Zak showed that for a secant defective manifold necessarily $N\le{n+2 \choose n}-1$ and that the Veronese variety $v_2(¶^n)$ is the only boundary case. Recently R. Mu$\tilde{\textrm{n}}$oz, J. C. Sierra, and L. E. Solá Conde classified secant defective varieties next to this extremal case in \cite{MSS}. In this paper, we will consider secant defective manifolds $X\subset¶^N$ of dimension $n$ with $N={n+2 \choose n}-1-ε$ for $ε\ge0$. First, we will prove that $X$ is a $LQEL$-manifold of type $δ=1$ for $ε\le n-2$ (see Theorem \ref{main_thm}) by showing that the tangential behavior of $X$ is good enough to apply Scorza lemma. Then we will completely describe the above manifolds by using the classification of conic-connected manifolds given in \cite{IR1}. Our method generalizes previous results in \cite{Z1,MSS}. |
| title | Classification of secant defective manifolds near the extremal case |
| topic | Algebraic Geometry 14MXX, 14NXX, 14M22 |
| url | https://arxiv.org/abs/1108.4264 |