  For $q\leq 3$ smooth plane algebraic curves $\mathcal{C}_i$ having simple
normal crossings, if the invariant logarithmic $2$-jet differential bundle
associated to $(\mathbb{P}^2(\mathbb{C}), \sum_{i=1}^q \mathcal{C}_i)$ has a
nonzero section vanishing on some ample divisor, then, for every algebraically
nondegenerate entire holomorphic curve
$f\colon\mathbb{C}\rightarrow\mathbb{P}^2(\mathbb{C})$, we have a Second Main
Theorem type estimate: \[ T_f(r) \leq c\sum_{i=1}^q\,N_f^{[1]}(r,\mathcal{C}_i)
+ o\big(T_f(r) \big)\parallel, \] where $T_f(r)$ and $N_f^{[1]}(r,C_i)$ stand
for the order function and the $1$--truncated counting functions in the
Nevanlinna theory, and where the constant $c=c(q,d_i)>0$ can be computed
explicitly. In particular, our result includes the case of $3$ conics in
$\mathbb{P}^2(\mathbb{C})$. Moreover, we provide some new results concerning
the algebraic degeneracy of certain complex surfaces, e.g., the complex
hyperbolicity of a very generic surface of degree $\geq 15$ in
$\mathbb{P}^3(\mathbb{C})$.
