Taiwanese Journal of Mathematics

Reducibility of the Hilbert Scheme of Smooth Curves and Families of Double Covers

Youngook Choi, Hristo Iliev, and Seonja Kim

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Let $\mathcal{I}_{d,g,r}$ be the union of irreducible components of the Hilbert scheme whose general points represent smooth irreducible complex curves of degree $d$ and genus $g$ in $\mathbb{P}^r$. Severi claimed in [15] that $\mathcal{I}_{d,g,r}$ is irreducible if $d \geq g+r$. His statement turned out to be correct for $r = 3$ and $4$, while for $r \geq 6$, counterexamples have been found by using families of $m$-sheeted covers of rational curves with $m \geq 3$. In this work we show the existence of an additional component of $\mathcal{I}_{d,g,r}$ whose general elements are double covers of curves of positive genus. In addition, we find upper bounds for the dimension of the possible components of $\mathcal{I}_{d,g,r}$.

Article information

Taiwanese J. Math., Volume 21, Number 3 (2017), 583-600.

First available in Project Euclid: 1 July 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14C05: Parametrization (Chow and Hilbert schemes)
Secondary: 14H10: Families, moduli (algebraic)

Hilbert scheme of curves Brill-Noether theory double covering


Choi, Youngook; Iliev, Hristo; Kim, Seonja. Reducibility of the Hilbert Scheme of Smooth Curves and Families of Double Covers. Taiwanese J. Math. 21 (2017), no. 3, 583--600. doi:10.11650/tjm/7839. https://projecteuclid.org/euclid.twjm/1498874608

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