Reference request: positive first chern class of tangent bundle implies anticanonical line bundle is ample

Question

I am searching for a reference (preferably with a proof) for the following result:

Let $X$ be a smooth projective curve, $T_X$ its tangent bundle, $K_X$ its canonical bundle. If $c_1(T_X) > 0$, then $K_X^{-1}$ is ample.

Answer 1

As discussed in the comments, one can prove the identity $c_1(\det E) = c_1(E)$ using the splitting principle. Since $K_X^{-1} = \det T_X$, we see that $$c_1(K_X^{-1}) = c_1(\det T_X) = c_1(T_X) > 0$$ and therefore $K_X^{-1}$ is ample. Here we have used the isomorphism $\det(E^*)^* \cong \det E$ to identify $K_X^{-1}$ with $\det(T_X)$ (since $K_X^{-1} = K_X^*$). Alternatively, we could use the identity $c_1(E^*) = -c_1(E)$ to reach the same conclusion: $$c_1(K_X^{-1}) = c_1(K_X^*) = -c_1(K_X) = -c_1(\det T^*_X) = -c_1(T^*_X) = c_1(T_X) > 0.$$