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We discuss criteria for a stable map of genus two and degree 4 to the projective plane to be smoothable, as an application of our modular desingularisation ofM   ‾    2  ,  nP   r   ,  dmain$\overline{\mathcal{M}}_{2, n}\left(\mathbb{P}^r, d\right)^{\text {main}}$   via logarithmic geometry and Gorenstein singularities.

A geographical study of M ‾ 2 P 2 , 4 main $\overline{\mathcal{M}}_2\left(\mathbb{P}^2, 4\right)^{\text {main }}$