Linear stability and rank two Clifford indices of algebraic curves with applications
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911146957602816 |
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| author | Bajravani, Ali Ortega, Angela |
| author_facet | Bajravani, Ali Ortega, Angela |
| contents | We prove that any vector bundle computing the rank-two Clifford index of a smooth projective algebraic curve is linearly semistable. We also identify conditions under which such bundles become linearly stable, thereby addressing a question posed by A. Castorena, G. H. Hitching and E. Luna in the rank-two case. Furthermore, we demostrate that in certain special cases, this property is equivalent to the (semi)stability of the associated Lazarsfeld-Mukai bundles. This yields a positive answer, in specific cases, to a generalized version of a conjecture proposed by Mistretta and Stoppino. We also study the moduli space $S_0(n,d,5)$ of generated $α$-stable coherent systems of type $(n,d,5)$ for small values of $α$ and $n=2,3$. We show that a general element of an irreducible component of $X \subseteq S_0(2,d,5)$ or $X \subseteq S_0(3,d,5)$ is linearly stable whenever $2δ_2 \leq d \leq \frac{3g}{2}$. As an application of this, we prove that Butler's conjecture holds non-trivially for coherent systems of type $(2,d,5)$ within the given range for $d$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_06149 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Linear stability and rank two Clifford indices of algebraic curves with applications Bajravani, Ali Ortega, Angela Algebraic Geometry 14H45, 14H51, 14H60 We prove that any vector bundle computing the rank-two Clifford index of a smooth projective algebraic curve is linearly semistable. We also identify conditions under which such bundles become linearly stable, thereby addressing a question posed by A. Castorena, G. H. Hitching and E. Luna in the rank-two case. Furthermore, we demostrate that in certain special cases, this property is equivalent to the (semi)stability of the associated Lazarsfeld-Mukai bundles. This yields a positive answer, in specific cases, to a generalized version of a conjecture proposed by Mistretta and Stoppino. We also study the moduli space $S_0(n,d,5)$ of generated $α$-stable coherent systems of type $(n,d,5)$ for small values of $α$ and $n=2,3$. We show that a general element of an irreducible component of $X \subseteq S_0(2,d,5)$ or $X \subseteq S_0(3,d,5)$ is linearly stable whenever $2δ_2 \leq d \leq \frac{3g}{2}$. As an application of this, we prove that Butler's conjecture holds non-trivially for coherent systems of type $(2,d,5)$ within the given range for $d$. |
| title | Linear stability and rank two Clifford indices of algebraic curves with applications |
| topic | Algebraic Geometry 14H45, 14H51, 14H60 |
| url | https://arxiv.org/abs/2509.06149 |