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$\mathbb{P}^5$
We denote by 
$\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth, irreducible, and non-degenerate curve of degree d and genus g in 
$\mathbb{P}^r.$ In this article, we study 
$\mathcal{H}_{15,g,5}$ for every possible genus g and determine when it is irreducible. We also study the moduli map 
$\mathcal{H}_{15,g,5}\rightarrow\mathcal{M}_g$ and several key properties such as gonality of a general element as well as characterizing smooth elements of each component.
