Let $M$ be a compact manifold with a metric $g$ and with a fixed spin structure $\chi$. Let $\lambda_1^+(g)$ be the first non-negative eigenvalue of the Dirac operator on $(M,g,\chi)$. We set $$\tau(M,\chi):= \sup \inf \lambda_1^+(g)$$ where the infimum runs over all metrics $g$ of volume $1$ in a conformal class $[g_0]$ on $M$ and where the supremum runs over all conformal classes $[g_0]$ on $M$. Let $(M^\#,\chi^\#)$ be obtained from $(M,\chi)$ by $0$-dimensional surgery. We prove that $$\tau(M^\#,\chi^\#)\geq \tau(M,\chi).$$ As a corollary we can calculate $\tau(M,\chi)$ for any Riemann surface $M$.
The spinorial $\tau$-invariant and 0-dimensional surgery