Segre invariants of principal bundles over a curve
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915955145179136 |
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| author | Hitching, George H. Zamora, Alfonso |
| author_facet | Hitching, George H. Zamora, Alfonso |
| contents | For a vector bundle $V$ over a curve $X$, the Segre invariant $s_n (V)$ encodes the maximal degree attained by rank $n$ subbundles of $V$. The functions $s_n$ define stratifications on moduli of $V$ which are well studied. Let $G$ be a connected reductive algebraic group, and $E \to X$ a principal $G$-bundle. For each parabolic subgroup $P \subset G$ there is a Segre number $s_P (E)$, generalising $s_n (V)$. We show that $s_P$ is semicontinuous in families of $G$-bundles, and thus defines stratifications on moduli spaces of $G$-bundles over $X$. We study the invariance properties of $s_P$, relating the behaviour of $s_P$ and $s_{ϕ(P)}$ for a surjective homomorphism $ϕ\colon G \to H$ and allowing us to compare the Segre stratifications for $G$ and $H$. Finally, we analyse the stratification for the Borel subgroup $B$ of ${\rm GL}_3$, identifying patterns in the geometry and proving, in particular, a sharp Hirschowitz-type bound on $s_B (E)$ for certain topological types. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_22465 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Segre invariants of principal bundles over a curve Hitching, George H. Zamora, Alfonso Algebraic Geometry 14H60, 14L35, 14D20 For a vector bundle $V$ over a curve $X$, the Segre invariant $s_n (V)$ encodes the maximal degree attained by rank $n$ subbundles of $V$. The functions $s_n$ define stratifications on moduli of $V$ which are well studied. Let $G$ be a connected reductive algebraic group, and $E \to X$ a principal $G$-bundle. For each parabolic subgroup $P \subset G$ there is a Segre number $s_P (E)$, generalising $s_n (V)$. We show that $s_P$ is semicontinuous in families of $G$-bundles, and thus defines stratifications on moduli spaces of $G$-bundles over $X$. We study the invariance properties of $s_P$, relating the behaviour of $s_P$ and $s_{ϕ(P)}$ for a surjective homomorphism $ϕ\colon G \to H$ and allowing us to compare the Segre stratifications for $G$ and $H$. Finally, we analyse the stratification for the Borel subgroup $B$ of ${\rm GL}_3$, identifying patterns in the geometry and proving, in particular, a sharp Hirschowitz-type bound on $s_B (E)$ for certain topological types. |
| title | Segre invariants of principal bundles over a curve |
| topic | Algebraic Geometry 14H60, 14L35, 14D20 |
| url | https://arxiv.org/abs/2604.22465 |