By L. Comtet

Although its identify, the reader won't locate during this ebook a scientific account of this large topic. sure classical facets were glided by, and the genuine name must be "Various questions of undemanding combina torial analysis". for example, we merely comment on the topic of graphs and configurations, yet there exists a truly broad and solid literature in this topic. For this we refer the reader to the bibliography on the finish of the amount. the real beginnings of combinatorial research (also known as combina tory research) coincide with the beginnings of likelihood thought within the seventeenth century. for approximately centuries it vanished as an self sustaining sub ject. however the improve of information, with an ever-increasing call for for configurations in addition to the appearance and improvement of pcs, have, past doubt, contributed to reinstating this topic after the sort of lengthy interval of negligence. for a very long time the purpose of combinatorial research used to be to count number the various methods of arranging items lower than given situations. consequently, a number of the conventional difficulties of study or geometry that are con cerned at a definite second with finite buildings, have a combinatorial personality. at the present time, combinatorial research can be appropriate to difficulties of lifestyles, estimation and structuration, like any different elements of mathema tics, yet completely forjinite units.

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X; to / equals '. (i) There exists a unique morphism « W Y T ! x/; t / on X T . x/ on X T ). Proof. x; t / 7! x; t / 7! OX / D OY in view of Lemma 1. y; to /, where y 2 Y , and the assumption (iii) of that lemma holds with s being the inclusion of Y To in X To . Finally, the assumption (iv)0 of Remark 13 is satisfied, since T is connected. By that remark, we thus have g D g ı s ı f on X T . Hence there 46 M. Brion exists a unique morphism « W Y T ! y; t /. x/ on X T . t u Remark 14. The preceding result has a nice interpretation when X is projective.

In view of Theorem 5, every nontrivial algebraic semigroup law on an irreducible curve S is commutative; by Proposition 17 again, it follows that S has an idempotent F -point whenever S and are defined over F . 4 Rigidity In this subsection, we obtain two rigidity results (both possibly known, but for which we could not locate adequate references) and we apply them to the study of endomorphisms of complete varieties. Our first result is a scheme-theoretic version of a classical rigidity lemma for irreducible varieties (see [8, Lem.

D 0 ; a contradiction. Thus, we must have n D 2, and we obtain a nonconstant morphism D 2 W S ! A1 , where the semigroup law on A1 is the multiplication. The image of contains 0 and a nonempty open subset U of the unit group Gm . Then U U D Gm and hence is surjective. e/ D 1. Then e is the desired nonzero idempotent. t u Remark 10. One may also deduce the above theorem from the description of algebraic semigroup structures on abelian varieties (Proposition 21), when the irreducible curve S is assumed to be nonsingular and nonrational.