Terracini loci and a codimension one Alexander-Hirschowitz theorem
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866908477309321216 |
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| author | Ballico, Edoardo Brambilla, Maria Chiara Fontanari, Claudio |
| author_facet | Ballico, Edoardo Brambilla, Maria Chiara Fontanari, Claudio |
| contents | The Terracini locus $\mathbb{T}(n, d; x)$ is the locus of all finite subsets $S$ of $ \mathbb{P}^n$ of cardinality $x$ such that $\langle S \rangle = \mathbb{P}^n$, $h^0(\mathcal{I}_{2S}(d)) > 0$, and $h^1(\mathcal{I}_{2S}(d)) > 0$. The celebrated Alexander-Hirschowitz Theorem classifies the triples $(n,d,x)$ for which $\dim\mathbb{T}(n, d; x)=xn$. Here we fully characterize the next step in the case $n=2$, namely, we prove that $\mathbb{T}(2,d;x)$ has at least one irreducible component of dimension $2x-1$ if and only if either $(d,x)\in\{(4,4),(4,6),$ $(5,6),(5,7),$ $(6,9),(6,10)\}$, or $d\ge 7$, $d\equiv 1,2 \pmod{3}$ and $x=(d+2)(d+1)/6$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_18751 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Terracini loci and a codimension one Alexander-Hirschowitz theorem Ballico, Edoardo Brambilla, Maria Chiara Fontanari, Claudio Algebraic Geometry Commutative Algebra 14C20, 14H50 The Terracini locus $\mathbb{T}(n, d; x)$ is the locus of all finite subsets $S$ of $ \mathbb{P}^n$ of cardinality $x$ such that $\langle S \rangle = \mathbb{P}^n$, $h^0(\mathcal{I}_{2S}(d)) > 0$, and $h^1(\mathcal{I}_{2S}(d)) > 0$. The celebrated Alexander-Hirschowitz Theorem classifies the triples $(n,d,x)$ for which $\dim\mathbb{T}(n, d; x)=xn$. Here we fully characterize the next step in the case $n=2$, namely, we prove that $\mathbb{T}(2,d;x)$ has at least one irreducible component of dimension $2x-1$ if and only if either $(d,x)\in\{(4,4),(4,6),$ $(5,6),(5,7),$ $(6,9),(6,10)\}$, or $d\ge 7$, $d\equiv 1,2 \pmod{3}$ and $x=(d+2)(d+1)/6$. |
| title | Terracini loci and a codimension one Alexander-Hirschowitz theorem |
| topic | Algebraic Geometry Commutative Algebra 14C20, 14H50 |
| url | https://arxiv.org/abs/2407.18751 |