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2021 The maximal genus of space curves in the range A
Edoardo Ballico, Philippe Ellia
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Albanian J. Math. 15(1): 10-38 (2021). DOI: 10.51286/albjm/1608313772

Abstract

Fix positive integers $d, m$ such that $\frac{m^2+4m+6}{6} \leq d \lt \frac{m^2+4m+6}{3}$ (the so-called Range A for space curves). Let $G(d, m)$ be the maximal genus of a smooth and connected degree $d$ curve $C \subset \mathbb P^3$ such that $h^0(\mathcal I_C(m-1)) = 0$. Here we prove that $G(d, m) = 1+(m-1)d -\binom{m+2}{3}$ if $m\ge 13.8\cdot 10^5$. The case $\frac{m^2+4m+6}{4} \leq d \lt \frac{m^2+4m+6}{3}$ was known by work of Fløystad [14, 15] and joint work of Ballico, Bolondi, Ellia, Mirò-Roig; see [2]. To prove the case $\frac{m^2+4m+6}{6} \leq d \lt \frac{m^2+4m+6}{4}$ we show that in this range for large $d$ every integer between $0$ and $1+(m-1)d -\binom{m+2}{3}$ is the genus of some degree $d$ smooth and connected curve $C \subset \mathbb P^3$ such that $h^0(\mathcal I_C(m-1)) = 0$.

Funding Statement

Partially supported by MIUR and GNSAGA of INdAM (Italy).

Citation

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Edoardo Ballico. Philippe Ellia. "The maximal genus of space curves in the range A." Albanian J. Math. 15 (1) 10 - 38, 2021. https://doi.org/10.51286/albjm/1608313772

Information

Published: 2021
First available in Project Euclid: 26 March 2021

Digital Object Identifier: 10.51286/albjm/1608313772

Subjects:
Primary: 14H51

Keywords: genus , Hilbert function , postulation , space curves

Rights: Copyright © 2021 Research Institute of Science and Technology (RISAT)

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Vol.15 • No. 1 • 2021
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