Limit linear series for curves not of compact type
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Brian Osserman
Abstract
We introduce a notion of limit linear series for nodal curves which are not of compact type. We give a construction of a moduli space of limit linear series, which works also in smoothing families, and we prove a corresponding specialization result. For a more restricted class of curves which simultaneously generalizes two-component curves and curves of compact type, we give an equivalent definition of limit linear series, which is visibly a generalization of the Eisenbud–Harris definition. Finally, for the same class of curves, we prove a smoothing theorem which constitutes an improvement over known results even in the compact-type case.
Funding source: National Security Agency
Award Identifier / Grant number: H98230-11-1-0159
Funding source: Simons Foundation
Award Identifier / Grant number: 279151
Funding statement: The author was partially supported by NSA grant H98230-11-1-0159 and Simons Foundation grant #279151 during the preparation of this work.
Acknowledgements
I would like to thank Eduardo Esteves for many helpful conversations, particularly in relation to chain structures and admissible multidegrees. I would also like to thank Frank Sottile for drawing my attention to [7], and Xiang He for helpful comments. Finally, I would like to thank the referee for a thorough and thoughtful reading.
References
[1] O. Amini and M. Baker, Linear series on metrized complexes of algebraic curves, Math. Ann. 362 (2015), no. 1–2, 55–106. 10.1007/s00208-014-1093-8Search in Google Scholar
[2] M. Baker and S. Norine, Riemann–Roch and Abel–Jacobi theory on a finite graph, Adv. Math. 215 (2007), no. 2, 766–788. 10.1016/j.aim.2007.04.012Search in Google Scholar
[3] F. Cools, J. Draisma, S. Payne and E. Robeva, A tropical proof of the Brill–Noether theorem, Adv. Math. 230 (2012), no. 2, 759–776. 10.1016/j.aim.2012.02.019Search in Google Scholar
[4] D. Eisenbud and J. Harris, Divisors on general curves and cuspidal rational curves, Invent. Math. 74 (1983), 371–418. 10.1007/BF01394242Search in Google Scholar
[5]
D. Eisenbud and J. Harris,
The Kodaira dimension of the moduli space of curves of genus
[6] E. Esteves and N. Medeiros, Limit canonical systems on curves with two components, Invent. Math. 149 (2002), no. 2, 267–338. 10.1007/s002220200211Search in Google Scholar
[7] L. García-Puente, N. Hein, C. Hillar, A. Martín del Campo, J. Ruffo, F. Sottile and Z. Teitler, The secant conjecture in the real Schubert calculus, Exp. Math. 21 (2012), no. 3, 252–265. 10.1080/10586458.2012.661323Search in Google Scholar
[8] D. Gieseker, Stable curves and special divisors: Petri’s conjecture, Invent. Math. 66 (1982), 251–275. 10.1007/BF01389394Search in Google Scholar
[9] U. Goertz, On the flatness of models of certain Shimura varieties of PEL-type, Math. Ann. 321 (2001), 689–727. 10.1007/s002080100250Search in Google Scholar
[10] P. Griffiths and J. Harris, On the variety of special linear systems on a general algebraic curve, Duke Math. J. 47 (1980), 233–272. 10.1215/S0012-7094-80-04717-1Search in Google Scholar
[11] J. Harris and D. Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982), 23–88. 10.1007/BF01393371Search in Google Scholar
[12] L. Mainò, Moduli space of enriched stable curves, Ph.D. thesis, Harvard University, 1998. Search in Google Scholar
[13] J. Murray and B. Osserman, Linked determinantal loci and limit linear series, Proc. Amer. Math. Soc. 144 (2016), no. 6, 2399–2410. 10.1090/proc/12965Search in Google Scholar
[14] B. Osserman, A limit linear series moduli scheme, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 4, 1165–1205. 10.5802/aif.2209Search in Google Scholar
[15] B. Osserman, Limit linear series moduli stacks in higher rank, preprint (2014), https://arxiv.org/abs/1405.2937. Search in Google Scholar
[16] B. Osserman, Relative dimension of morphisms and dimension for algebraic stacks, J. Algebra 437 (2015), 52–78. 10.1016/j.jalgebra.2015.04.022Search in Google Scholar
[17] B. Osserman, Dimension counts for limit linear series on curves not of compact type, Math. Z. 284 (2016), no. 1–2, 69–93. 10.1007/s00209-016-1646-5Search in Google Scholar
[18] B. Osserman, Limit linear series and the Amini–Baker construction, in preparation. 10.1007/s00209-018-2170-6Search in Google Scholar
[19] S. Sternberg, Dynamical systems, Dover Publications, Mineola 2010. Search in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Homotopy techniques for tensor decomposition and perfect identifiability
- Schottky groups acting on homogeneous rational manifolds
- Limit linear series for curves not of compact type
- Uniform congruence counting for Schottky semigroups in SL2(𝐙)
- The dualizing sheaf on first-order deformations of toric surface singularities
- Embedded minimal surfaces of finite topology
- Spectral gap characterization of full type III factors
- On the universal cover and the fundamental group of an RCD*(K,N)-space
- Calabi–Yau and fractional Calabi–Yau categories
Articles in the same Issue
- Frontmatter
- Homotopy techniques for tensor decomposition and perfect identifiability
- Schottky groups acting on homogeneous rational manifolds
- Limit linear series for curves not of compact type
- Uniform congruence counting for Schottky semigroups in SL2(𝐙)
- The dualizing sheaf on first-order deformations of toric surface singularities
- Embedded minimal surfaces of finite topology
- Spectral gap characterization of full type III factors
- On the universal cover and the fundamental group of an RCD*(K,N)-space
- Calabi–Yau and fractional Calabi–Yau categories
