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In mathematics, the **de Franchis theorem** is one of a number of closely related statements applying to compact Riemann surfaces, or, more generally, algebraic curves, *X* and *Y*, in the case of genus *g* > 1. The simplest is that the automorphism group of *X* is finite (see though Hurwitz's automorphisms theorem). More generally,

- the set of non-constant morphisms from
*X*to*Y*is finite; - fixing
*X*, for all but a finite number of such*Y*, there is no non-constant morphism from*X*to*Y*.

These results are named for Michele De Franchis (1875–1946). It is sometimes referenced as the De Franchis-Severi theorem. It was used in an important way by Gerd Faltings to prove the Mordell conjecture.

- M. De Franchis:
*Un teorema sulle involuzioni irrazionali*, Rend. Circ. Mat Palermo 36 (1913), 368