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
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
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