On the singular loci of higher secant varieties of Veronese embeddings
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arXiv
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866916833020346368 |
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| author | Furukawa, Katsuhisa Han, Kangjin |
| author_facet | Furukawa, Katsuhisa Han, Kangjin |
| contents | The $k$-th secant variety of a projective variety $X \subset \mathbb{P}^N$, denoted by $σ_k(X)$, is defined to be the closure of the union of $(k-1)$-planes spanned by $k$ points on $X$. In this paper, we examine the $k$-th secant variety $σ_k(v_d(\mathbb{P}^n)) \subset \mathbb{P}^N$ of the image of the $d$-uple Veronese embedding $v_d$ of $\mathbb{P}^n$ to $\mathbb{P}^N$ with $N=\binom{n+d}{d}-1$, and focus on the singular locus of $σ_k(v_d(\mathbb{P}^n))$, which is only known for $k\le3$.
To study the singularity for arbitrary $k,d,n$, we define \emph{the $m$-subsecant locus} of $σ_k(v_d(\mathbb{P}^n))$ to be the union of $σ_k(v_d(\mathbb{P}^m))$ with any $m$-plane $\mathbb{P}^m \subset \mathbb{P}^n$. By investigating the projective geometry of moving embedded tangent spaces along subvarieties and using known results on the secant defectivity and the identifiability of symmetric tensors, we determine whether the $m$-subsecant locus is contained in the singular locus of $σ_k(v_d(\mathbb{P}^n))$ or not.
Depending on the value of $k$, these subsecant loci show an interesting trichotomy between generic smoothness, non-trivial singularity, and trivial singularity. In many cases, they can be used as a new source for the singularity of the $k$-th secant variety of $v_d(\mathbb{P}^n)$ other than the trivial one, the $(k-1)$-th secant variety of $v_d(\mathbb{P}^n)$. We also consider the case of the $4$-th secant variety of $v_d(\mathbb{P}^n)$ by applying main results and computing conormal space via a certain type of Young flattening. Finally, we present some generalizations and discussions for further developments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_03254 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On the singular loci of higher secant varieties of Veronese embeddings Furukawa, Katsuhisa Han, Kangjin Algebraic Geometry 14N05, 14M12, 14J60, 15A21, 15A69 The $k$-th secant variety of a projective variety $X \subset \mathbb{P}^N$, denoted by $σ_k(X)$, is defined to be the closure of the union of $(k-1)$-planes spanned by $k$ points on $X$. In this paper, we examine the $k$-th secant variety $σ_k(v_d(\mathbb{P}^n)) \subset \mathbb{P}^N$ of the image of the $d$-uple Veronese embedding $v_d$ of $\mathbb{P}^n$ to $\mathbb{P}^N$ with $N=\binom{n+d}{d}-1$, and focus on the singular locus of $σ_k(v_d(\mathbb{P}^n))$, which is only known for $k\le3$. To study the singularity for arbitrary $k,d,n$, we define \emph{the $m$-subsecant locus} of $σ_k(v_d(\mathbb{P}^n))$ to be the union of $σ_k(v_d(\mathbb{P}^m))$ with any $m$-plane $\mathbb{P}^m \subset \mathbb{P}^n$. By investigating the projective geometry of moving embedded tangent spaces along subvarieties and using known results on the secant defectivity and the identifiability of symmetric tensors, we determine whether the $m$-subsecant locus is contained in the singular locus of $σ_k(v_d(\mathbb{P}^n))$ or not. Depending on the value of $k$, these subsecant loci show an interesting trichotomy between generic smoothness, non-trivial singularity, and trivial singularity. In many cases, they can be used as a new source for the singularity of the $k$-th secant variety of $v_d(\mathbb{P}^n)$ other than the trivial one, the $(k-1)$-th secant variety of $v_d(\mathbb{P}^n)$. We also consider the case of the $4$-th secant variety of $v_d(\mathbb{P}^n)$ by applying main results and computing conormal space via a certain type of Young flattening. Finally, we present some generalizations and discussions for further developments. |
| title | On the singular loci of higher secant varieties of Veronese embeddings |
| topic | Algebraic Geometry 14N05, 14M12, 14J60, 15A21, 15A69 |
| url | https://arxiv.org/abs/2111.03254 |