An algebraic lifting invariant of Ellenberg, Venkatesh, and Westerland

We define and prove basic properties of a lifting invariant of curves over an algebraically closed field k with a map to the projective line $${\mathbb {P}}^1_{k}$$ that was introduced by Ellenberg, Venkatesh, and Westerland. In other work of Liu, Zureick-Brown, and the author, this lifting invariant, and its connection to components of Hurwitz schemes, is applied to help understand the average number of unramified extensions of function fields with a given Galois group.