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Progress in the theory of complex algebraic curves. (English) Zbl 0703.14016
The authors describe some important recent progress in the theory of algebraic curves. After a short sketch of the fundamental notions, culminating in the Riemann-Roch and the Brill-Noether theorems, they devote the remainder of the paper to moduli spaces. Riemann had already considered the set \({\mathcal M}_ g\) of isomorphism classes of smooth complex projective curves of genus \(g,\) and, by a remarkable computation, had shown that, for \(g\geq 2\) this set was “described” by \(3g-3\) complex parameters.
The modern theory starts with Teichmüller who defined a natural topology on \({\mathcal M}_ g\), and Bers, in 1970, proved that \({\mathcal M}_ g\) was homeomorphic to an open ball in \({\mathbb{C}}^{3g-3}\). Then, using D. Mumford’s “Geometric invariant theory” (1965; Zbl 0147.393), it is possible to define on \({\mathcal M}_ g\) a natural structure of an irreducible algebraic variety defined over \({\mathbb{Q}}\), as well as a “compactification” \(\bar {\mathcal M}_ g\) of that variety, which consists in adding to \({\mathcal M}_ g\) the isomorphism classes of curves having only ordinary double points as singularities, and having no smooth rational components meeting other components in less than three points. The recent research is concerned with properties of the algebraic variety \({\mathcal M}_ g\); the main problem is to determine if \({\mathcal M}_ g\) is unirational. This was stated by Severi for \(g\leq 10\), using explicit descriptions of \({\mathcal M}_ g\), which the authors quote for \(g\leq 4\); very recent work also prove unirationality for 11\(\leq g\leq 13\) and \(g=15\). But using cohomology and the Grothendieck-Riemann-Roch theorem, it is now proved that for \(g\geq 23\), \({\mathcal M}_ g\) is not unirational.
Reviewer: J.Dieudonné

MSC:
14H10 Families, moduli of curves (algebraic)
14-03 History of algebraic geometry
01A60 History of mathematics in the 20th century
14M20 Rational and unirational varieties
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