  Let $k$ be a field. Let $X/k$ be a stable curve whose geometric irreducible
components are smooth rational curves. Taking Stein factorization of its
normalization, we get a conic. We show the conic is non-split in certain cases.
As an application, we show for $g\geq3$, the period and index of the universal
genus $g$ curve both equal to $2g-2$.
