On a Theorem of Torelli

Andreotti, Aldo

doi:10.2307/2372835

Cited by 100 publications

(67 citation statements)

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“…Notice that the method of proof of Theorem 13.11 is analogous to the method of Andreotti in [1]. Our final theorem also derives from the techniques and results of that same work.…”

Section: The Gherardelli-todd Isomorphismmentioning

confidence: 84%

“…The geometric aspects of our proofs of (0.11) and (0.12) were motivated by Andreotti's proof of the Torelli theorem for curves [1]. His arguments are based on the interplay between the geometry of the canonical mapping of the curve into projective space and the Gauss mapping on the theta-divisor.…”

Section: ) (G -K)! (G -K)-timesmentioning

confidence: 99%

“…Let Djo be the double curve of Sv, j 1, 2. Then g(D10) g(D20) so that by the classical Torelli theorem [1], D10 D20. Thus a canonical embedding of D10 into P3 can differ from one for D20 by at most a linear automorphism of P3.…”

Section: Proof For (T T') E DI J(t T') (L( To + L#(t T')) Where mentioning

confidence: 99%

See 2 more Smart Citations

The Intermediate Jacobian of the Cubic Threefold

Clemens¹,

Griffiths²

1972

The Annals of Mathematics

596

689

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“…Notice that the method of proof of Theorem 13.11 is analogous to the method of Andreotti in [1]. Our final theorem also derives from the techniques and results of that same work.…”

Section: The Gherardelli-todd Isomorphismmentioning

confidence: 84%

Section: ) (G -K)! (G -K)-timesmentioning

confidence: 99%

See 1 more Smart Citation

The Intermediate Jacobian of the Cubic Threefold

Clemens¹,

Griffiths²

1972

The Annals of Mathematics

596

689

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“…This phase began in 1955 with the notes [41 ] and [42] in which I gave prescriptions for finding sets (each containing 3g -3 elements) of periods of normal abelian integrals of first kind serving as local moduli near a given nonhyperelliptic surface (in fact, local coordinates on T° near suitable points covering the surface class) and analogous sets of 2g -1 elements each serving as local coordinates on the hyper elliptic locus of T° near a point thereof. The motivating idea behind introducing the periods was an attempt to refine an important theorem of Torelli (circa 1912: for some recent treatments see [8], [32] and [55]) to the effect that two surfaces with the same period matrices are conformally equivalent (more precision in §2). Torelli's theorem furnishes a beautiful set of global moduli and gives rise (see §2) to a second naturally defined covering of M°, the Torelli space, which provides a natural setting for them, the only trouble being that the essential periods are superfluous in number, g(g + l)/2, in contrast to 3g -3.…”

Section: Introduction and Historymentioning

confidence: 99%

A transcendental view of the space of algebraic Riemann surfaces

Rauch¹

1965

Bull. Amer. Math. Soc.

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The subject of my lecture is not a large-scale and general theory but rather a special problem in conformai mapping. By way of justification for bringing it to the attention of a broader audience I may cite, first of all, its deep nature and honorable origin-the problem originated in Riemann's memoir [46] of 1857 on algebraic functions, and after the passage of a century some definitive insight into it is only now emerging as a result of the collective efforts of a group of mathematicians. Secondly, it is intimately bound up with a variety of topics of more general interest: conformai mapping with particular reference to extremal problems, several complex variables, and partial differential equations, on the one hand; topology and algebraic geometry, on the other and, in particular aspects, even group representations and arithmetic. Introduction and history.Briefly put, and in modern paraphrase, Riemann observed: (a) that for compact (algebraic) Riemann surfaces of genus g = 0, i.e., simply connected, one has the analogue of the Riemann mapping theorem, that is, all such surfaces not only admit topological maps on one another but also conformai maps, in particular, on the Riemann sphere (all maps of surfaces in this paper are assumed to be homeomorphisms without further mention) ; (b) for g>0 it is no longer true that the existence of a map of one surface on another (equivalent to their having the same genus) implies the existence of a conformai map; and (c) as a partial compensation for (b) and substitute for (a) in case g>0, if one considers the conformai equivalence classes obtained by identifying surfaces which admit mutual conformai maps, these classes form not an isolated point as in (a) but in some sense or other a continuum of finite dimension, in fact, of dimension one for g= 1 and 3g -3 for g^2. Riemann called a set of parameters for the continuum moduli (Riemann and his immediate successors often spoke cryptically of the moduli, as though some particular set of moduli were implicitly distinguished-this has not been borne out by the modern research to be described below).

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“…Using in turn the projection formula, (1), (5), (6), and (7), we get Wr-Wi = ir(Xr-(S + Xi)) s (2g -1 -r)Wr+i --ir(Ar+i).…”

Section: Ir(12pi)ewi If and Only If Xx~°> Where 6 = P-f-qi+ • • • +Q"mentioning

confidence: 99%