Conics associated with totally degenerate curves

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 \(2g-2\).