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April 2022 Space curves on surfaces with ordinary singularities
Mengyuan Zhang
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Illinois J. Math. 66(1): 1-29 (April 2022). DOI: 10.1215/00192082-9753971


We show that smooth curves in the same biliaison class on a hypersurface in P3 with ordinary singularities are linearly equivalent. We compute the invariants h0(IC(d)), h1(IC(d)), and h1(OC(d)) of a curve C on such a surface X in terms of the cohomologies of divisors on the normalization of X. We then study general projections in P3 of curves lying on the rational normal scroll S(a,b)Pa+b+1. If we vary the curves in a linear system on S(a,b) as well as the projections, we obtain a family of curves in P3. We compute the dimension of the space of deformations of these curves in P3 as well as the dimension of the family. We show that the difference is a linear function in a and b which does not depend on the linear system. Finally, we classify maximal rank curves on ruled cubic surfaces in P3. We prove that the general projections of all but finitely many classes of projectively normal curves on S(1,2)P4 fail to have maximal rank in P3. These give infinitely many classes of counter-examples to a question of Hartshorne.


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Mengyuan Zhang. "Space curves on surfaces with ordinary singularities." Illinois J. Math. 66 (1) 1 - 29, April 2022.


Received: 21 October 2019; Revised: 20 January 2022; Published: April 2022
First available in Project Euclid: 20 February 2022

Digital Object Identifier: 10.1215/00192082-9753971

Primary: 14H50
Secondary: 14Q05 , 32S25

Rights: Copyright © 2022 by the University of Illinois at Urbana–Champaign


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Vol.66 • No. 1 • April 2022
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