# The differential of the Gauss map of the Theta Divisor of a quartic plane curve.

Let's suppose we have a non-hyperelliptic Riemann surface $X$ of genus $3$ (without lost of generality, embedded in $\mathbb{P}^2$ as a smooth plane curve) and we consider its Theta Divisor. Is well known that the Gauss Map of the Theta Divisor can be identified with the map which associates to the divisor $P+Q$ on $X$ the line in $\mathbb{P}^2$ spanned by those two points, and to the divisor $2P$ the tangent line to $X$ at the point $P$.

What about the differential of this map? What are the points of the Theta divisor for which the differential is not injective? Do they have a geometrical description?