Let $\pi:X_{\epsilon} \rightarrow \Delta$ be a family of (say smooth) projective plane curves parametrized by $\Delta:=\operatorname{Spec}(k[\epsilon])$, and let $X=X_0$ be the closed fiber. Suppose that $X_\epsilon$ is given by a polynomial $f(x,y,z;\epsilon)$ homogeneus in $x,y,z$.

Let $\phi=\operatorname{ks}(\pi)=\operatorname{ks}(\partial/\partial\epsilon) \in H^1(X,T_X)=H^0(X,K_X)^{*}$ be the Kodaira-Spencer image of the above family.

• Is it possible to characterize $\phi$ concretely in terms of the polynomial $f$?

If you want, feel free to restrict to the hyperelliptic case:

$f(x,y,1;\epsilon):=y^2-\Pi_{i=1}^{2g+2}(x-\lambda_i(\epsilon))$,

(which I think has to be desingularized though)

in which case a basis of $H^0(X,K_X)$ is given by $\frac{x^{k}}{y}dx$ for $k=0 \cdots g-1$.