In algebraic geometry, the Zeuthen–Segre invariant "I" is an invariant of a projective surface found in a complex projective space which was introduced by Zeuthen (1871) and rediscovered by Corrado Segre (1896).
The invariant "I" is defined to be "d" – 4"g" – "b" if the surface has a pencil of curves, non-singular of genus "g" except for "d" curves with 1 ordinary node, and with "b" base points where the curves are non-singular and transverse. 
Alexander (1914) showed that the Zeuthen–Segre invariant "I" is χ–4, where χ is the topological Euler–Poincaré characteristic introduced by Poincaré (1895), which is equal to the Chern number "c"2 of the surface.