A positive proportion of locally soluble hyperelliptic curves over $\mathbb{Q}$ have no point over any odd degree extension

A hyperelliptic curve over $\mathbb{Q}$ is called ``locally soluble'' if it has a point over every completion of $\mathbb{Q}$. In this paper, we prove that a positive proportion of hyperelliptic curves over $\mathbb{Q}$ of genus $g\geq 1$ are locally soluble but have no points over any odd degree extension of $\mathbb{Q}$. We also obtain a number of related results. For example, we prove that for any fixed odd integer $k > 0$, the proportion of locally soluble hyperelliptic curves over $\mathbb{Q}$ of genus $g$ having no points over any odd degree extension of $\mathbb{Q}$ of degree at most $k$ tends to $1$ as $g$ tends to infinity. We also show that the failures of the Hasse principle in these cases are explained by the Brauer-Manin obstruction. Our methods involve a detailed study of the geometry of pencils of quadrics over a general field of characteristic not equal to $2$, together with suitable arguments from the geometry of numbers.