On the prime ideals of higher secant varieties of Veronese embeddings of small degrees
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917792346800128 |
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| author | Furukawa, Katsuhisa Han, Kangjin |
| author_facet | Furukawa, Katsuhisa Han, Kangjin |
| contents | In this paper, we study minimal generators of the (saturated) defining ideal of $σ_k(v_d(\mathbb{P}^n))$ in $\mathbb{P}^{N}$ with ${N=\binom{n+d}{d}-1}$, the $k$-secant variety of $d$-uple Veronese embedding of projective $n$-space, of a relatively small degree. We first show that the prime ideal $I(σ_4(v_3(\mathbb{P}^3)))$ can be minimally generated by 36 homogeneous polynomials of degree $5$. It implies that $σ_4(v_3(\mathbb{P}^3)) \subset \mathbb{P}^{19}$ is a del Pezzo $4$-secant variety (i.e., $\mathrm{deg}(σ_4(v_3(\mathbb{P}^3))) = 105$ and the sectional genus $π(σ_4(v_3(\mathbb{P}^3))) = 316$) and provides a new example of an arithmetically Gorenstein variety of codimension $4$. As an application, we decide non-singularity of a certain locus in $σ_4(v_3(\mathbb{P}^3))$. By inheritance, generators of $I(σ_4(v_3(\mathbb{P}^n)))$ are also obtained for any $n \geq 3$.
We also propose a procedure to compute the first non-trivial degree piece $I(σ_k(v_d(\mathbb{P}^n)))_{k+1}$ for a general $k$-th secant case, in terms of prolongation and weight space decomposition, based on the method used for $σ_4(v_3(\mathbb{P}^3))$ and treat a few more cases of $k$-secant varieties of the Veronese embedding of a relatively small degree in the end. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_00652 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the prime ideals of higher secant varieties of Veronese embeddings of small degrees Furukawa, Katsuhisa Han, Kangjin Algebraic Geometry Commutative Algebra 13P05, 13C14, 14N05, 14N25, 15A69 In this paper, we study minimal generators of the (saturated) defining ideal of $σ_k(v_d(\mathbb{P}^n))$ in $\mathbb{P}^{N}$ with ${N=\binom{n+d}{d}-1}$, the $k$-secant variety of $d$-uple Veronese embedding of projective $n$-space, of a relatively small degree. We first show that the prime ideal $I(σ_4(v_3(\mathbb{P}^3)))$ can be minimally generated by 36 homogeneous polynomials of degree $5$. It implies that $σ_4(v_3(\mathbb{P}^3)) \subset \mathbb{P}^{19}$ is a del Pezzo $4$-secant variety (i.e., $\mathrm{deg}(σ_4(v_3(\mathbb{P}^3))) = 105$ and the sectional genus $π(σ_4(v_3(\mathbb{P}^3))) = 316$) and provides a new example of an arithmetically Gorenstein variety of codimension $4$. As an application, we decide non-singularity of a certain locus in $σ_4(v_3(\mathbb{P}^3))$. By inheritance, generators of $I(σ_4(v_3(\mathbb{P}^n)))$ are also obtained for any $n \geq 3$. We also propose a procedure to compute the first non-trivial degree piece $I(σ_k(v_d(\mathbb{P}^n)))_{k+1}$ for a general $k$-th secant case, in terms of prolongation and weight space decomposition, based on the method used for $σ_4(v_3(\mathbb{P}^3))$ and treat a few more cases of $k$-secant varieties of the Veronese embedding of a relatively small degree in the end. |
| title | On the prime ideals of higher secant varieties of Veronese embeddings of small degrees |
| topic | Algebraic Geometry Commutative Algebra 13P05, 13C14, 14N05, 14N25, 15A69 |
| url | https://arxiv.org/abs/2410.00652 |