Abstract
This paper is the second of a two-part series by the author devoted to the following fundamental problem in the theory of algebraic curves in projective space: Which reducible curves arise as limits of smooth curves of general moduli? Special cases of this question studied by Sernesi (Sernesi, Edoardo (1984) On the existence of certain families of curves. Invent Math 75(1): 488 25-57 ), Ballico (Ballico, Edoardo (2012) Embeddings of general curves in projective spaces: the range of the quadrics. Lith Math J 52(2): 134-137 ), and others have been critical in the resolution of many problems in the theory of algebraic curves over the past half century. In this paper, we give sharp bounds on this problem for space curves, when the nodes are general points and the components are general in moduli. We also systematically study a variant in projective spaces of arbitrary dimension when the nodes are general in a hyperplane. The results given here significantly extend those cases established in the previous paper in this series (Eric Larson, Constructing reducible Brill-Noether curves, To appear in documentamathematica, arxiv:1603.02301), as well as those cases established by Sernesi (Sernesi, Edoardo (1984) On the existence of certain families of curves. Invent Math 75(1): 488 25-57 ), Ballico (Ballico, Edoardo (2012) Embeddings of general curves in projective spaces: the range of the quadrics. Lith Math J 52(2): 134-137 ) , and others. As explained in (Eric Larson, Degenerations of curves in projective space and the maximal rank conjecture, arXiv:1809.05980), the reducible curves constructed in this paper also play a critical role in the author’s proof of the maximal rank conjecture in a subsequent paper (Eric Larson Degenerations of curves in projective space and the maximal rank conjecture, arXiv:1809.05980).
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Acknowledgements
The author would like to thank Joe Harris for his guidance throughout this research, as well as other members of the Harvard and MIT mathematics departments for helpful conversations. The author would also like to acknowledge the generous support both of the Fannie and John Hertz Foundation, the Department of Defense (NDSEG fellowship), and the National Science Foundation (MSPRF grant DMS-1802908). Finally, the author would like to thank the anonymous referee for many helpful suggestions.
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Larson, E. Constructing reducible Brill–Noether curves II. manuscripta math. 172, 75–88 (2023). https://doi.org/10.1007/s00229-022-01408-9
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DOI: https://doi.org/10.1007/s00229-022-01408-9