The complete classification of isotopy classes of degree three symplectic curves in $\mathbb{CP}^2$ via a novel algebraic theory of braid monodromy

Abstract

We develop a new algebraic theory of positive braids and conjugacy classes in the braid group $B_3$. We use our theory to establish a complete classification of isotopy classes of degree three symplectic curves in $\mathbb{CP}^2$ with only $A_n$-singularities for $n\geq 1$ (an $A_n$-singularity is locally modelled by the equation $z^2 = w^n$) independent of Gromov's theory of pseudoholomorphic curves. We show that if $C$ and $C'$ are degree three symplectic curves in $\mathbb{CP}^2$ with the same numbers of $A_n$-singularities for each $n\geq 1$, then $C$ is isotopic to $C'$. Furthermore, our theory furnishes a single method of proof that independently establishes and unifies several fundamental classification results on degree three symplectic curves in $\mathbb{CP}^2$. In particular, we prove using our theory: (1) there is a unique isotopy class of degree three smooth symplectic curves in $\mathbb{CP}^2$ (a result due to Sikorav), (2) the number of nodes is a complete invariant of the isotopy class of a degree three nodal symplectic curve in $\mathbb{CP}^2$ (the case of irreducible nodal curves is due to Shevchishin and the case of reducible nodal curves is due to Golla-Starkston), and (3) there is a unique isotopy class of degree three cuspidal symplectic curves in $\mathbb{CP}^2$ (a generalization of a result due to Ohta-Ono). The present work represents the first step toward resolving the symplectic isotopy conjecture using purely algebraic techniques in the theory of braid groups. Finally, we independently establish a complete classification of genus one Lefschetz fibrations over $\mathbb{S}^2$ (a result due to Moishezon-Livne).