It was introduced by Émile Picard for complex surfaces in his book Picard & Simart (1897), and extended to higher dimensions by Solomon Lefschetz (1924).
Pierre Deligne and Nicholas Katz (1973) extended Picard–Lefschetz theory to varieties over more general fields, and Deligne used this generalization in his proof of the Weil conjectures.
The Picard–Lefschetz formula describes the monodromy at a critical point.
Also suppose that all critical points are non-degenerate and lie in different fibers, and have images x1,...,xn in P1.
The fundamental group π1(P1 – {x1, ..., xn}, x) is generated by loops wi going around the points xi, and to each point xi there is a vanishing cycle in the homology Hk(Yx) of the fiber at x.
The monodromy action of π1(P1 – {x1, ..., xn}, x) on Hk(Yx) is described as follows by the Picard–Lefschetz formula.
The monodromy action of a generator wi of the fundamental group on
∈ Hk(Yx) is given by where δi is the vanishing cycle of xi.
This formula appears implicitly for k = 2 (without the explicit coefficients of the vanishing cycles δi) in Picard & Simart (1897, p.95).
Lefschetz (1924, chapters II, V) gave the explicit formula in all dimensions.
Consider the projective family of hyperelliptic curves of genus
of a generic curve is the matrix we can easily compute the Picard-Lefschetz formula around a degeneration on