Counting generalized Jenkins–Strebel differentials
Abstract We study the combinatorial geometry of “lattice” Jenkins–Strebel differentials with simple zeroes and simple poles on $$\mathbb{C }\!\mathrm{P }^1$$ and of the corresponding counting functions. Developing the results of Kontsevich (Commun Math Phys 147:1–23, 1992) we evaluate the leading te...
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| Autor*in: |
Athreya, Jayadev S. [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2013 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer Science+Business Media Dordrecht 2013 |
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| Übergeordnetes Werk: |
Enthalten in: Geometriae dedicata - Springer Netherlands, 1972, 170(2013), 1 vom: 18. Juni, Seite 195-217 |
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| Übergeordnetes Werk: |
volume:170 ; year:2013 ; number:1 ; day:18 ; month:06 ; pages:195-217 |
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| DOI / URN: |
10.1007/s10711-013-9877-7 |
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OLC2039062599 |
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| 520 | |a Abstract We study the combinatorial geometry of “lattice” Jenkins–Strebel differentials with simple zeroes and simple poles on $$\mathbb{C }\!\mathrm{P }^1$$ and of the corresponding counting functions. Developing the results of Kontsevich (Commun Math Phys 147:1–23, 1992) we evaluate the leading term of the symmetric polynomial counting the number of such “lattice” Jenkins–Strebel differentials having all zeroes on a single singular layer. This allows us to express the number of general “lattice” Jenkins–Strebel differentials as an appropriate weighted sum over decorated trees. The problem of counting Jenkins–Strebel differentials is equivalent to the problem of counting pillowcase covers, which serve as integer points in appropriate local coordinates on strata of moduli spaces of meromorphic quadratic differentials. This allows us to relate our counting problem to calculations of volumes of these strata . A very explicit expression for the volume of any stratum of meromorphic quadratic differentials recently obtained by the authors (Athreya et al. 2012) leads to an interesting combinatorial identity for our sums over trees. | ||
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Counting generalized Jenkins–Strebel differentials |
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Abstract We study the combinatorial geometry of “lattice” Jenkins–Strebel differentials with simple zeroes and simple poles on $$\mathbb{C }\!\mathrm{P }^1$$ and of the corresponding counting functions. Developing the results of Kontsevich (Commun Math Phys 147:1–23, 1992) we evaluate the leading term of the symmetric polynomial counting the number of such “lattice” Jenkins–Strebel differentials having all zeroes on a single singular layer. This allows us to express the number of general “lattice” Jenkins–Strebel differentials as an appropriate weighted sum over decorated trees. The problem of counting Jenkins–Strebel differentials is equivalent to the problem of counting pillowcase covers, which serve as integer points in appropriate local coordinates on strata of moduli spaces of meromorphic quadratic differentials. This allows us to relate our counting problem to calculations of volumes of these strata . A very explicit expression for the volume of any stratum of meromorphic quadratic differentials recently obtained by the authors (Athreya et al. 2012) leads to an interesting combinatorial identity for our sums over trees. © Springer Science+Business Media Dordrecht 2013 |
| abstractGer |
Abstract We study the combinatorial geometry of “lattice” Jenkins–Strebel differentials with simple zeroes and simple poles on $$\mathbb{C }\!\mathrm{P }^1$$ and of the corresponding counting functions. Developing the results of Kontsevich (Commun Math Phys 147:1–23, 1992) we evaluate the leading term of the symmetric polynomial counting the number of such “lattice” Jenkins–Strebel differentials having all zeroes on a single singular layer. This allows us to express the number of general “lattice” Jenkins–Strebel differentials as an appropriate weighted sum over decorated trees. The problem of counting Jenkins–Strebel differentials is equivalent to the problem of counting pillowcase covers, which serve as integer points in appropriate local coordinates on strata of moduli spaces of meromorphic quadratic differentials. This allows us to relate our counting problem to calculations of volumes of these strata . A very explicit expression for the volume of any stratum of meromorphic quadratic differentials recently obtained by the authors (Athreya et al. 2012) leads to an interesting combinatorial identity for our sums over trees. © Springer Science+Business Media Dordrecht 2013 |
| abstract_unstemmed |
Abstract We study the combinatorial geometry of “lattice” Jenkins–Strebel differentials with simple zeroes and simple poles on $$\mathbb{C }\!\mathrm{P }^1$$ and of the corresponding counting functions. Developing the results of Kontsevich (Commun Math Phys 147:1–23, 1992) we evaluate the leading term of the symmetric polynomial counting the number of such “lattice” Jenkins–Strebel differentials having all zeroes on a single singular layer. This allows us to express the number of general “lattice” Jenkins–Strebel differentials as an appropriate weighted sum over decorated trees. The problem of counting Jenkins–Strebel differentials is equivalent to the problem of counting pillowcase covers, which serve as integer points in appropriate local coordinates on strata of moduli spaces of meromorphic quadratic differentials. This allows us to relate our counting problem to calculations of volumes of these strata . A very explicit expression for the volume of any stratum of meromorphic quadratic differentials recently obtained by the authors (Athreya et al. 2012) leads to an interesting combinatorial identity for our sums over trees. © Springer Science+Business Media Dordrecht 2013 |
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| title_short |
Counting generalized Jenkins–Strebel differentials |
| url |
https://doi.org/10.1007/s10711-013-9877-7 |
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| author2 |
Eskin, Alex Zorich, Anton |
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Eskin, Alex Zorich, Anton |
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| doi_str |
10.1007/s10711-013-9877-7 |
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