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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2508.08426 |
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| _version_ | 1866915441343987712 |
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| author | Abenda, Simonetta Grinevich, Petr G. |
| author_facet | Abenda, Simonetta Grinevich, Petr G. |
| contents | ${\mathtt{MM}}$-curves are rational degenerations of ${\mathtt{M}}$-curves, i.e. they are maximal Mumford in the sense that they posses $g$ tropical cycles and exactly $g+1$ real ovals, where $g$ is the arithmetic genus. For rational curves the ``naive'' definition of divisors as formal sums of points requires a refinement. In the finite-gap theory of KP II equation the real regular solutions correspond to the Dubrovin-Natanzon (DN) divisors on ${\mathtt{M}}$-curves. In the case of real regular multiline KP II solitons, it was shown by the authors that for any given solution there exists a normalization time such that the spectral data are smooth DN divisor on ${\mathtt{MM}}$-curve.
However, to show that DN divisors parameterize the; full positroid cell, it is necessary to fix the normalization time and consider both smooth and non-smooth divisors. In this paper we start such an investigation, and show that on ${\mathtt{MM}}$-curves whose dual graphs are trivalent Le-graphs of totally positive Schubert cells, the construction of non-smooth DN divisors requires combinations of just two basic types of blow-ups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_08426 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dubrovin-Natanzon divisors on MM-curves Abenda, Simonetta Grinevich, Petr G. Algebraic Geometry Mathematical Physics Combinatorics Exactly Solvable and Integrable Systems 14C20, 14H70, 14P99, 37K10, 05C90 ${\mathtt{MM}}$-curves are rational degenerations of ${\mathtt{M}}$-curves, i.e. they are maximal Mumford in the sense that they posses $g$ tropical cycles and exactly $g+1$ real ovals, where $g$ is the arithmetic genus. For rational curves the ``naive'' definition of divisors as formal sums of points requires a refinement. In the finite-gap theory of KP II equation the real regular solutions correspond to the Dubrovin-Natanzon (DN) divisors on ${\mathtt{M}}$-curves. In the case of real regular multiline KP II solitons, it was shown by the authors that for any given solution there exists a normalization time such that the spectral data are smooth DN divisor on ${\mathtt{MM}}$-curve. However, to show that DN divisors parameterize the; full positroid cell, it is necessary to fix the normalization time and consider both smooth and non-smooth divisors. In this paper we start such an investigation, and show that on ${\mathtt{MM}}$-curves whose dual graphs are trivalent Le-graphs of totally positive Schubert cells, the construction of non-smooth DN divisors requires combinations of just two basic types of blow-ups. |
| title | Dubrovin-Natanzon divisors on MM-curves |
| topic | Algebraic Geometry Mathematical Physics Combinatorics Exactly Solvable and Integrable Systems 14C20, 14H70, 14P99, 37K10, 05C90 |
| url | https://arxiv.org/abs/2508.08426 |