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| In [[mathematics]], the concept of [[abelian variety]] is the higher-dimensional generalization of the [[elliptic curve]]. The '''equations defining abelian varieties''' are a topic of study because every abelian variety is a [[projective variety]]. In dimension ''d'' ≥ 2, however, it is no longer as straightforward to discuss such equations.
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| There is a large classical literature on this question, which in a reformulation is, for [[complex algebraic geometry]], a question of describing relations between [[theta function]]s. The modern geometric treatment now refers to some basic papers of [[David Mumford]], from 1966 to 1967, which reformulated that theory in terms from abstract algebraic geometry valid over general [[field (mathematics)|field]]s.
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| ==Complete intersections==
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| The only 'easy' cases are those for ''d'' = 1, for an elliptic curve with linear span the projective plane or projective 3-space. In the plane, every elliptic curve is given by a cubic curve. In ''P''<sup>3</sup>, an elliptic curve can be obtained as the intersection of two [[quadric]]s.
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| In general abelian varieties are not [[complete intersection]]s. [[Computer algebra]] techniques are now able to have some impact on the direct handling of equations for small values of ''d'' > 1.
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| ==Kummer surfaces==
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| The interest in nineteenth century geometry in the [[Kummer surface]] came in part from the way a [[quartic surface]] represented a quotient of an abelian variety with ''d'' = 2, by the group of order 2 of automorphisms generated by ''x'' → −''x'' on the abelian variety.
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| ==General case==
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| Mumford defined a [[theta group]] associated to an [[invertible sheaf]] ''L'' on an abelian variety ''A''. This is a group of self-automorphisms of ''L'', and is a finite analogue of the [[Heisenberg group]]. The primary results are on the action of the theta group on the [[global section]]s of ''L''. When ''L'' is [[very ample]], the [[linear representation]] can be described, by means of the structure of the theta group. In fact the theta group is abstractly a simple type of [[nilpotent group]], a [[Group extension%23Central extension|central extension]] of a group of torsion points on ''A'', and the extension is known (it is in effect given by the [[Weil pairing]]). There is a uniqueness result for irreducible linear representations of the theta group with given [[central character]], or in other words an analogue of the [[Stone–von Neumann theorem]]. (It is assumed for this that the characteristic of the field of coefficients doesn't divide the order of the theta group.)
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| Mumford showed how this abstract algebraic formulation could account for the classical theory of theta functions with [[theta characteristic]]s, as being the case where the theta group was an extension of the two-torsion of ''A''.
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| An innovation in this area is to use the [[Mukai–Fourier transform]].
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| ==The coordinate ring==
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| The goal of the theory is to prove results on the [[homogeneous coordinate ring]] of the embedded abelian variety ''A'', that is, set in a projective space according to a very ample ''L'' and its global sections. The [[graded commutative ring]] that is formed by the direct sum of the global sections of the
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| :<math>L^n,\ </math>
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| meaning the ''n''-fold [[tensor product]] of itself, is represented as the [[quotient ring]] of a [[polynomial algebra]] by a [[homogeneous ideal]] ''I''. The graded parts of ''I'' have been the subject of intense study.
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| Quadratic relations were provided by [[Bernhard Riemann]]. '''Koizumi's theorem''' states the third power of an ample line bundle is [[normally generated]]. The '''Mumford–Kempf theorem''' states that the fourth power of an ample line bundle is quadratically presented. For a base field of [[characteristic zero]], Giuseppe Pareschi proved a result including these (as the cases ''p'' = 0, 1) which had been conjectured by Lazarsfeld: let ''L'' be an ample line bundle on an abelian variety ''A''. If ''n'' ≥ ''p'' + 3, then the ''n''-th tensor power of ''L'' satisfies [[Homogeneous coordinate ring#Projective normality|condition ''N''<sub>p</sub>]].<ref>Giuseppe Pareschi, ''Syzygies of Abelian Varieties'', Journal of the American Mathematical Society, Vol. 13, No. 3 (Jul., 2000), pp. 651–664.</ref> Further results have been proved by Pareschi and Popa, including previous work in the field.<ref>Giuseppe Pareschi, Minhea Popa, ''Regularity on abelian varieties II: basic results on linear series and defining equations'', J. Alg. Geom. 13 (2004), 167–193; http://www.math.uic.edu/~mpopa/papers/abv2.pdf</ref>
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| ==See also==
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| * [[Timeline of abelian varieties]]
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| * [[Horrocks–Mumford bundle]]
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| ==References==
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| * [[David Mumford]], ''On the equations defining abelian varieties I'' Invent. Math., 1 (1966) pp. 287–354
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| *____, ''On the equations defining abelian varieties II–III'' Invent. Math. , 3 (1967) pp. 71–135; 215–244
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| *____, ''Abelian varieties'' (1974)
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| *[[Jun-ichi Igusa]], ''Theta functions'' (1972)
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| {{Reflist}}
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| ==Further reading==
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| *[[David Mumford]], ''Selected papers on the classification of varieties and moduli spaces'', editorial comment by G. Kempf and H. Lange, pp. 293–5
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| [[Category:Abelian varieties]]
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