en>John of Reading: Typo and General fixing, replaced: the are amongst → they are amongst using AWB

2011-07-01T16:25:04Z

Typo and General fixing, replaced: the are amongst → they are amongst using AWB

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In [[algebraic geometry]], the '''homogeneous coordinate ring''' ''R'' of an [[algebraic variety]] ''V'' given as a [[subvariety]] of [[projective space]] of a given dimension ''N'' is by definition the [[quotient ring]]

:''R'' = ''K''[''X''0, ''X''1, ''X''2, ..., ''X''''N'']/''I''

where ''I'' is the [[homogeneous ideal]] defining ''V'', ''K'' is the [[algebraically closed field]] over which ''V'' is defined, and

:''K''[''X''0, ''X''1, ''X''2, ..., ''X''''N'']

is the [[polynomial ring]] in ''N'' + 1 variables ''X''i. The polynomial ring is therefore the homogeneous coordinate ring of the projective space itself, and the variables are the [[homogeneous coordinates]], for a given choice of basis (in the [[vector space]] underlying the projective space). The choice of basis means this definition is not intrinsic, but it can be made so by using the [[symmetric algebra]].

==Formulation==

Since ''V'' is assumed to be a variety, and so an [[irreducible algebraic set]], the ideal ''I'' can be chosen to be a [[prime ideal]], and so ''R'' is an [[integral domain]]. The same definition can be used for general homogeneous ideals, but the resulting coordinate rings may then contain non-zero [[nilpotent element]]s and other [[divisors of zero]]. From the point of view of [[scheme theory]] these cases may be dealt with on the same footing by means of the [[Proj construction]].

The correspondence between homogeneous ideals ''I'' and varieties is bijective for ideals not containing the ideal ''J'' generated by all the ''X''''i'', which corresponds to the empty set because not all homogeneous coordinates can vanish at a point of projective space. This correspondence is known as [[projective Nullstellensatz]].

==Resolutions and syzygies==

In application of [[homological algebra]] techniques to algebraic geometry, it has been traditional since [[David Hilbert]] (though modern terminology is different) to apply [[free resolution]]s of ''R'', considered as a [[graded module]] over the polynomial ring. This yields information about [[Syzygy (mathematics)|syzygies]], namely relations between generators of the ideal ''I''. In a classical perspective, such generators are simply the equations one writes down to define ''V''. If ''V'' is a [[hypersurface]] there need only be one equation, and for [[complete intersection]]s the number of equations can be taken as the codimension; but the general projective variety has no defining set of equations that is so transparent. Detailed studies, for example of [[canonical curve]]s and the [[equations defining abelian varieties]], show the geometric interest of systematic techniques to handle these cases. The subject also grew out of [[elimination theory]] in its classical form, in which reduction modulo ''I'' is supposed to become an algorithmic process (now handled by [[Gröbner bases]] in practice).

There are for general reasons free resolutions of ''R'' as graded module over ''K''[''X''0, ''X''1, ''X''2, ..., ''X''''N'']. A resolution is defined as ''minimal'' if the image in each module morphism of [[free module]]s

:φ:''F''''i'' → ''F''''i'' − 1

in the resolution lies in ''JF''''i'' − 1. As a consequence of [[Nakayama's lemma]] φ then takes a given basis in ''F''''i'' to a minimal set of generators in ''F''''i'' − 1. The concept of ''minimal free resolution'' is well-defined in a strong sense, in that such a resolution is unique ([[up to]] isomorphism of [[chain complex]]es) and occurs as a [[direct summand]] in any free resolution. This property of being intrinsic to ''R'' allows the definition of the '''graded Betti numbers''', namely the β''i, j'' which are the number of grade-''j'' images coming from ''F''''i'' (more precisely, by thinking of φ as a matrix of homogeneous polynomials, the count of entries of that homogeneous degree incremented by the gradings acquired inductively from the right). In other words weights in all the free modules may be inferred from the resolution, and the graded Betti numbers count the number of generators of a given weight in a given module of the resolution. The discussion of these invariants of ''V'' in a given projective embedding is a research area, even in the case of curves.<ref>[[David Eisenbud]], ''The Geometry of Syzygies'', (2005, ISBN 978-0-387-22215-8), pp. 5–8.</ref>

There are examples where the minimal free resolution is known explicitly. For a [[rational normal curve]] it is an [[Eagon–Northcott complex]]. For [[elliptic curve]]s in projective space the resolution may be constructed as a [[mapping cone of complexes|mapping cone]] of Eagon–Northcott complexes.<ref>Eisenbud, Ch. 6.</ref>

==Regularity==

The [[Castelnuovo–Mumford regularity]] may be read off the minimum resolution of the ideal ''I'' defining the projective variety. In terms of the imputed "shifts" ''a''''i'', ''j'' in the ''i''-th module ''F''''i'', it is the maximum over ''i'' of the ''a''''i'', ''j'' − ''i''; it is therefore small when the shifts increase only by increments of 1 as we move to the left in the resolution (linear syzygies only).<ref>Eisenbud, Ch. 4.</ref>

==Projective normality==

The variety ''V'' in its projective embedding is '''projectively normal''' if ''R'' is [[integrally closed]]. This condition implies that ''V'' is a [[normal variety]], but not conversely: the property of projective normality is not independent of the projective embedding, as is shown by the example of a rational quartic curve in three dimensions.<ref>Robin Hartshorne, ''Algebraic Geometry'' (1977), p. 23.</ref> Another equivalent condition is in terms of the [[linear system of divisors]] on ''V'' cut out by the [[tautological line bundle]] ''L'' on projective space, and its ''d''-th powers for ''d'' = 1, 2, 3, ... ; when ''V'' is [[non-singular]], it is projectively normal if and only if each such linear system is a [[complete linear system]].<ref>Hartshorne, p. 159.</ref> In a more geometric way one can think of ''L'' as the [[Serre twist sheaf]] ''O''(1) on projective space, and use it to twist the structure sheaf ''O''''V'' ''k'' times, for any ''k''. Then ''V'' is called ''k''-normal if the global sections of ''O''(''k'') map surjectively to those of ''O''''V''(''k''), for a given ''k''; if ''V'' is 1-normal it is called '''linearly normal''', and projective normality is the condition that ''V'' is ''k''-normal for all ''k'' ≥ 1. [[Linear normality]] may be said geometrically: ''V'' as projective variety cannot be obtained by an isomorphic [[linear projection]] from a projective space of higher dimension, except in the trivial way of lying in a proper linear subspace. Projective normality may similarly be translated, by using enough [[Veronese mapping]]s to reduce it to conditions of linear normality.

Looking at the issue from the point of view of a given [[very ample line bundle]] giving rise to the projective embedding of ''V'', such a line bundle ([[invertible sheaf]]) is said to be '''normally generated''' if ''V'' as embedded is projectively normal. Projective normality is the first condition ''N''0 of a sequence of conditions defined by Green and Lazarsfeld. For this

:<math>\bigoplus_{d=0}^\infty H^0(V, L^d)</math>

is considered as graded module over the homogeneous coordinate ring of the projective space, and a minimal free resolution taken. Condition ''N''p applied to the first ''p'' graded Betti numbers, requiring they vanish when ''j'' > ''i'' + 1.<ref>See e.g. Elena Rubei, ''On Syzygies of Abelian Varieties'', Transactions of the American Mathematical Society, Vol. 352, No. 6 (Jun., 2000), pp. 2569–2579.</ref> For curves Green showed that condition ''N''''p'' is satisfied when deg(''L'') ≥ 2''g'' + 1 + ''p'', which for ''p'' = 0 was a classical result of [[Guido Castelnuovo]].<ref>Giuseppe Pareschi, ''Syzygies of Abelian Varieties'', Journal of the American Mathematical Society, Vol. 13, No. 3 (Jul., 2000), pp. 651–664.</ref>

==See also==
*[[Projective variety]]
*[[Hilbert polynomial]]

==Notes==
{{Reflist}}

==References==
*[[Oscar Zariski]] and [[Pierre Samuel]], ''Commutative Algebra'' Vol. II (1960), pp. 168–172.

[[Category:Algebraic varieties]]

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en>John of Reading: Typo and General fixing, replaced: the are amongst → they are amongst using AWB