Brill-Noether loci
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866912077153566720 |
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| author | Bigas, Montserrat Teixidor i |
| author_facet | Bigas, Montserrat Teixidor i |
| contents | Brill-Noether loci ${\mathcal M}^r_{g,d}$ are those subsets of the moduli space ${\mathcal M}_g$ determined by the existence of a linear series of degree $d$ and dimension $r$. By looking at non-singular curves in a neighborhood of a special chain of elliptic curves, we provide a new proof of the non-emptiness of the Brill-Noether loci when the expected codimension satisfies $-g+r+1\le ρ(g,r,d)\le 0$ and prove that for a generic point of a component of this locus, the Petri map is onto. As an application, we show that Brill-Noether loci of the same codimension are distinct when the codimension is not too large, substantially generalizing the known result in codimensions 1 and 2. We also provide a new technique for checking that Brill-Noether loci are not included in each other. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_10581 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Brill-Noether loci Bigas, Montserrat Teixidor i Algebraic Geometry 14H10, 14H51 Brill-Noether loci ${\mathcal M}^r_{g,d}$ are those subsets of the moduli space ${\mathcal M}_g$ determined by the existence of a linear series of degree $d$ and dimension $r$. By looking at non-singular curves in a neighborhood of a special chain of elliptic curves, we provide a new proof of the non-emptiness of the Brill-Noether loci when the expected codimension satisfies $-g+r+1\le ρ(g,r,d)\le 0$ and prove that for a generic point of a component of this locus, the Petri map is onto. As an application, we show that Brill-Noether loci of the same codimension are distinct when the codimension is not too large, substantially generalizing the known result in codimensions 1 and 2. We also provide a new technique for checking that Brill-Noether loci are not included in each other. |
| title | Brill-Noether loci |
| topic | Algebraic Geometry 14H10, 14H51 |
| url | https://arxiv.org/abs/2308.10581 |