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2017 Reducibility of the Hilbert Scheme of Smooth Curves and Families of Double Covers
Youngook Choi, Hristo Iliev, Seonja Kim
Taiwanese J. Math. 21(3): 583-600 (2017). DOI: 10.11650/tjm/7839


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 $\mathcal{I}_{d,g,r}$.


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Youngook Choi. Hristo Iliev. Seonja Kim. "Reducibility of the Hilbert Scheme of Smooth Curves and Families of Double Covers." Taiwanese J. Math. 21 (3) 583 - 600, 2017.


Published: 2017
First available in Project Euclid: 1 July 2017

zbMATH: 06871333
MathSciNet: MR3661382
Digital Object Identifier: 10.11650/tjm/7839

Primary: 14C05
Secondary: 14H10

Rights: Copyright © 2017 The Mathematical Society of the Republic of China


Vol.21 • No. 3 • 2017
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