Karol Palka


The Coolidge-Nagata conjecture


The Coolidge-Nagata conjecture asserts that every complex planar rational curve which has only cusps as singularities is Cremona equivalent to a line. Using the Logarithmic Minimal Model Program for boundaries with half-integral coefficients we showed [1] that for curves violating this conjecture the process of minimalization has length at most one, and hence the log resolution divisor is small (in particular there are at most two cusps). Together with M. Koras we now established the Coolidge-Nagata conjecture in the remaining cases. I will discuss our proof.

[1] K. Palka, The Coolidge–Nagata conjecture, part I., Adv. Math. 267 (2014), 1–43. http://www.sciencedirect.com/science/article/pii/S0001870814002783