We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational curves with irreducible singularities (rational cuspidal curves) in the complex projective plane. We prove that every such curve is isotopic to a complex curve in degrees up to five, and for curves with one singularity whose link is a torus knot. Classification results of symplectic isotopy classes rely on pseudo-holomorphic curves together with a symplectic version of birational geometry of log pairs and techniques from four-dimensional topology.

F. Cukierman asked whether or not for every smooth real plane curve $X\subset \mathbb{P}^{2}$ of even degree $d\geqslant 2$ there exists a real line $L\subset \mathbb{P}^{2}$ such $X\cap L$ has no real points. We show that the answer is yes if $d=2$ or 4 and no if $n\geqslant 6$.

The splitting number of a plane irreducible curve for a Galois cover is effective in distinguishing the embedded topology of plane curves. In this paper, we define the connected number of a plane curve (possibly reducible) for a Galois cover, which is similar to the splitting number. By using the connected number, we distinguish the embedded topology of Artal arrangements of degree $b\,\ge \,4$, where an Artal arrangement of degree $b$ is a plane curve consisting of one smooth curve of degree $b$ and three of its total inflectional tangents.

Recently, differential geometric properties of embedded projective varieties have gained increasing interest. In this note, we consider plane algebraic curves equipped with the Fubini–Study metric from${\mirrored P}$$_2$(${\mirrored C}$) and give an estimate for the diameter in terms of the degree, initiated in a paper by F. A. Bogomolov.