# Origins and history of branched covering

During my research on branched coverings of the projective plane, I am interested to know the origins and history of branched coverings of the projective plane and the projective line, together with the rise of the concept of "branch point" or "branch curve" (resp. "ramification points" or "ramification curve"). It seems that the modern conception was developed by Zariski and Segre in the 20s and 30s of the 20th century, and the example of Zariski-Segre of the branch curve of degree 6 (having 6 cusps on a conic) of a cubic surface projected generically to the projective plane, is well known. But were there other mathematicians before them who helped conceptualize this concept?

Thank you, Thomas

The theory of branched (or ramified) coverings has its origins in continuation of analytic functions and the attempts to find maximal analytic continuations of a given function. However, certain complex functions, e.g. $f(z) =z^{1/2}$ are multi-valued in certain subdomains of the complex plane, so when trying to continue along the closed curve one might arrive at another branch of the multivalued function, not the original one, so the valued do not match. The idea of Riemann surfaces offers a geometric way to deal with this problem by introducing a Riemann surface as the natural domain of $z^{1/2}$ and similarly for other troublesome functions. One can picture the Riemann surface for $z^{1/2}$ as two sheets coming together at $z=0$. The Riemann surfaces appeared in Bernhard Riemann's inaugural thesis in 1851, http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Grund/Grund.pdf, and are considered the first examples of covering spaces.
• Thanks! these are coverings of the complex line, right? as $f$ operates from $\mathbb C^1$ to $\mathbb C^1$. When the research of branch curves (i.e. of coverings of the projective complex plane) began? – Thomas Aug 14 '16 at 8:59
• Yes, a (compact) Riemann surface associated to an algebraic function is a covering of the projective line (=Riemann sphere), and conversely, if a meromorphic function $f: M \to \hat{mathbb{C}}$ defined on a compact Riemann surface is an $n$-fold covering of the Riemann sphere, then this function is algebraic. Compact Riemann surfaces are exactly projective algebraic manifolds of complex dimension one. I do not have good historical information about higher dimensions, though. – Margaret Friedland Aug 14 '16 at 18:12