A note on surfaces in ${\mathbb{C}\mathbb{P}}^2$ and ${\mathbb{C}\mathbb{P}}^2\mathrm{\#}{\mathbb{C}\mathbb{P}}^2$

In this brief note, we investigate the ${\mathbb{C}\mathbb{P}}^2$-genus of knots, i.e. the least genus of a smooth, compact, orientable surface in ${\mathbb{C}\mathbb{P}}^2\setminus \mathring{B^4}$ bounded by a knot in $S^3$. We show that this quantity is unbounded, unlike its topological counterpart. We also investigate the ${\mathbb{C}\mathbb{P}}^2$-genus of torus knots. We apply these results to improve the minimal genus bound for some homology classes in ${\mathbb{C}\mathbb{P}}^2\mathrm{\#}{\mathbb{C}\mathbb{P}}^2$.