First introduction to Projective Toric Varieties

Chapter 1

Projective toric varieties are a type of possiblysingular complex manifolds indexed by easycombinatorial data having to do with poles andzeroes of meromorphic functions.

It is maybe easiest to introduce the combinatorialdata first. We start with an abstract lattice ofsome rank n which we may take to be justthe set Zn, whose elements we could denoteon paper by columns of length n with integerentries.

Then any finite set S ⊂ Zn describes a projectivetoric variety, if it is not empty.

The easiest way of describing the singular manifoldis to say that S has an “affine structure.” Althoughit is not always possible to add elements of S toobtain other elements of S, it is certainly possibleto determine the truth or falsehood of any additiveequation involving elements of S. Because we won’tbe caring about the choice of origin 0 ⊂ Zn wewill restrict attention to homogeneous equations, suchas the equation a+ b+ c = d+ e+ f among six elementsof S ⊂ Zn. This particular equation is homogenousof degree three, but any degree is allowed.

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We say that an element x ∈ Zn is in the convex hull ofS if an equation holds such as

x+ x+ x = a+ b+ c

for elements a, b, c ∈ S not necessarily distinct. Andwe say that S is convex if it is equal to its convexhull.

Ordinarily one only considers convex subsets S ⊂ Zn

and if one uses a non-convex subset, the resultingcomplex manifold is not called ‘toric’ by convention,though there is a well-defined process called ‘normalization’that can be applied to the singular manifold itselfto get back to the one corresponding to the convex hull.

A function h : S → C to the complex numbers is called,let us say, an affine map if it preserves the affinestructures, where we use multiplication in C. This meansthat whenever we have an equation in S like

a+ b+ c = d+ e+ f

we should also have

h(a)h(b)h(c) = h(d)h(e)h(f).

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The projective toric variety corresponding to S isdefined (usually only when S is convex) to bethe equivalence classes of affine maps S → Cwhich are not identically zero, under the relationthat two maps h, h′ are considered equivalent ifthere is a nonzero complex number λ such that

h = λh′ .

Example. If the elements of S are affinely linearlyindependent (meaning that if one is considered to bethe origin the others are linearly independent), thenthere are no homogeneous equations among elementsof S except tautologies like a+b=b+a etcetera.This means that the affine maps h : S → C are merelyall functions. Labelling the elements of S as

x0, ..., xm

where m is the dimension of the affine spanof S ⊂ Zn then we see that the projective toricvariety is just all ratios [x0 : ... : xm] which isthe set of points in the projective space Pm ofdimension m.

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Example. The previous example is a littlerestrictive if we require S to be convex, becausethere are not many convex affinely independentsets. They are, I think, just the affine bases of(cosets of) summands of Zn.

However, if we ever expand a convex set S , simplytaking the multiples ks, for s ∈ S where k is a fixedinteger, and take the convex hull, weare in a situation where anyaffine map from the vertex set kS to C extendsuniquely to an affine map from the convex hull.

So the process of expanding S by an integer and takingthe convex hull does not affect the projective toric variety.

And it follows

Example. If S is an affine basis of a summandof the lattice and k is a nonzero integer, thenthe convex hull of kS defines the projective toricvariety which is a projective space.

In a certain sense these are the universal examples,because when there are relations among elementsof S it just means that some of thethe functions S → C are not allowed, andwe get a closed subset of projective spaceinsteadof the whole projective space.

For simplicity say S spans Zn affinely.

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The affine maps S → C which sendno element of S to zero extend uniquelyto affine maps

Zn → C×

where C× is the group of nonzerocomplex numbers under multiplication.

Instead of modding out by the action of nonzeroscalars, we can equivalently here justpass to the subset of affine maps sending 0 to 1.These are just the ordinary group homomorphismsZn → C× .

This set is bijective with (C×)n and it isa group under multiplication. It is called a“torus” because each factor of C×

contains the unit circle, and a cartesian product ofn copies of a circle is sometimes called an n

dimensional torus.

But here we call the larger group of realdimension 2n a torus also.

And thus we see that any projective toricvariety contains a torus.

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Example.

Let’s look at the four points in Z2 which have entriesof 0 and 1. These are the four columns of the matrix

(0 0 1 10 1 0 1

).

We call these a, b, c, d and the only nontrivial relation

is

a+ d = b+ c.

We visualize the letters a, b, c, d as the four cornerof a square (with coordinates as given by the columnsw chose),

a b

c d

The various afine functions h merely assignnumbers to a, b, c, d and the affine requirmenttogether with the nonzero requirement justmeans be the entries of a matrix of rank one.

Thus we can see the elements a, b, c, d ∈ S asordinary variables, and the arrangement as a square,the positions we have writen them on the page,require us to substitute only such numbers asyield a rank one matrix.

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So the variety is rank one matrices modulomultiplication by nonzero scalars. And wesee that such a thing is uniquly determinedby just the ratios [a : b] and [a : c] so thatour variety is just the cartesian product oftwo copies of one dimensional projective space.

We could have seen this more easily by workingcomponent-by-component. Our set S was afterall only the cartesian product of two copies of0, 1 ⊂ Z .

However, here we have constructed the same surfaceas the solution set of the homogeneous equationad = bc and it is a subset of projective three-space.

Note that the square diagram above is uniquely a lineardegeneration of a tetrahedron; the linear degeneration isthe defining constraint for the surface within projectivespace.

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Chapter 2 – Divisors

In this section I’ll use define some wordswhich I won’t end up using, so if somethingdoesn’t make sense, don’t worry and just keepreading.

Now let’s talk about divisors. An irreducibledivisor on a complex manifold means a possiblysingular codimension one submanifoldwhich is not a union of smaller ones.

A divisor means either a finite union ofirreducible divisors, or, more generally,a finite sequence of irreducible divisorsto which are attached integers.

A meromorphic function on a normal complexprojective variety has a divisor of poles andzeroes, and by convention we count polesas being negative zeroes, so that if a meromorphicfunction could have only zeroes, its divisorwould have only positive numbers attachedto its irreducible components. But in factthis never happens. A meromorphic functionwithout any poles would be defined everywhereon the complex projective variety, but thereis no such function except constants.

One way around this difficulty is to considerinstead of only functions, rather to consider

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the more general things which are sections ofline bundles. For any locally principal divisor, onecan make a holomorphic line bundle such thatyour divisor is exactly equal to the divisor ofzeroes (and poles) of a meromorphic section.

Because you can always multiply a meromorphicsection of a line bundle by a meromorphic function,then once you can obtain your divisor from a linebundle, you can equally well obtain any divisor whichdiffers from your divisor by the divisor of a meromorphicfunction.

Divisors coming from meromorphic functions(meromorphic sections of the trivial bundle) arecalled “principal divisors” and two divisors whichdiffer only by a principal divisor are called“linearly equivalent.”

One can easily show by this correspondencethat isomorphism types of holomorphic linebundles are naturally bijective with the a groupof divisors (the locally principal divisors) moduloprincipal divisors. If the variety is smooth, alldivisors are locally principal.

Smoothness exercise. Show that the manifold is smoothif and only if S is convex and each vertex ofthe convex hull together with the verticesconnected by an edge form an affine lattice basis.

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Although it is actually easier, because it isabstract, we won’t here talk about line bundles,but only about divisors.

If we choose any codimension-one face of ourconvex set S, it will of course be a convexsubset on its own, and will define a codimnsionone subvariety.

The way this works is that a face S0 ⊂ S

corresponds to the classes mod scalars ofaffine maps S → C which send all but theelements of S0 to zero.

Exercise. Show that the set of elements not sent tozero under any affine map S → C is either emptyor a face of some dimension.

If the codimension one faces of S are F1, ..., Fu thenthe divisor in which each Fi is given coefficient −1is denoted

−F1 − F2 − ...− Fu

and it is called a “canonical divisor” becauseit is the divisor of (zeros and) poles of a meromorphicdifferential n forms where n is thedimension of the variety (we call it n consistentlywith the superscript in Zn as if S spans affinelyour variety has dimension n).

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Actually, I’ve used the same letter Fi for thecodimension one face of S as I have for thecorresponding codimension one subset of ourvariety – the latter of course is all the affinemaps which are zero except on Fi.

Anyway, with this abuse of notation accepted,the only divisors we ever have to worry aboutare going to be integer linear combinationsof the Fi.

Again, this is something you don’t yet need to know,but the torus acts on the variety and there are no torusinvariant divisors except linear combinations of the Fi.

And every divisor is linearly equivalent to atorus invariant divisor.

And this means we won’t go wrong if we just pretendthat there are no divisors except which we make byassigning integers to the faces of S.

If we view S or perhaps its real convex hull as acell complex, then what we are talking aboutis exactly a cellular n− 1 chain.

Let me give you the next thing to visualize withoutexplaining it. What I want you to visualize is thatinstead of assinging integers to the faces of S, weare going to move the faces in and out.

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If you think of the faces as infinite hyperplanesin Zn, then as these hyperplanes are translated fromone position to another in Rn , they collide at certaintimes with points of Zn. Or, if we think of thehyperplanes as just being subsets of Zn, the translatesactually are cosets of Zn modulo whichever translatecontains 0, and the cosets are elements of an infinitecyclic quotient group.

Whenever I speak of ‘how far to move’ a faceof S I am going to mean using counting in theinfinite cyclic quotient group that has to do withthat face.

Now, it is best to start by choosing an element of S tobe the origin of the lattice, and having our movablehyperplanes F1, ..., Fu chosen as being the ones meetingat the origin. To construct S, we will move eachone in the originally chosen direction (outward) somenumber of steps. This gives a positive integer to assignto each Fi , and if we call these integers e1, ..., euthen we will write down the divisor e1F1 + ...+ euFu

This is an effective divisor because all ei are positive.The hyperplanes Fi each moved out ei steps providethe faces of the convex set S. This choice of the ei,or any other which defines a finite subset of the latticedefining the same projective variety, makese1F1 + ...+ euFu be a very ample divisor.

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I should have said this earlier, but if we choose anytwo points of Zn , say x, y, then for each point hof our variety, the ratio h(x)/h(y) is well definedas long as h(y) 6= 0.

It is convenient to take y to be the origin in Zn andthen if you recall when we restrict h to the torusas no element is sent to zero, instead of reducing moduloscalar multiplication it is better to assume h(0) = 1.Then our chosen element x ∈ Zn corresponds to thefunction

h 7→ h(x)

and this is something very familiar. For, justas the torus is group homomorphisms Zn → C×

we can recover the group Zn as analytic grouphomomorphisms (C×)n → C× .

Each lattice element x ∈ Zn already corresponds toan analytic function on the torus, and one which isa group homomorphism (a “character” of the torus).

And when we take y to be the origin of Zn the functionsending h to the well defined ratio h(x)/h(y) , whenwe restrict to the torus in our variety, is nothing butx itself, viewed as a character of the torus.

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Now I am going to just state something, withoutgetting us lost in any details. I said already that theconvex polyhedron we started with, because we neededto move the faces out in order that it should haveany volume at all, has positive integers assigned to itsfaces. The particular integers depended on a choiceof origin of the lattice.

With respect to the same origin, all the latticepoints contained in the polyhedron correspond to particularmeromorphic functions, and being inside the polyhedron(or on the boundary) means that all of these functionshave poles no worse than the prescribed divisor

e1F1 + ...+ euFu

This means that we can reinterpret our chosen convexset S as being the torus characters which have divisorof poles no worse than as prescribed by that divisor.

Now, each lattice point q also has a principal divisor.The origin of the lattice has the divisor zero. Startingwith all the hyperplanes in Zn intersectingat the origin, if we choose any other lattice point q, thenumber of steps (with outwards counting positively)each face has to move so that all meet atthe point q, counting steps in the infinitecyclic quotient group modulo that hyperplane,defines a divisor a1F1 + ...+ auFu, and this isthe divisor of zeroes and poles of q.

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The fact that not all of the ai can be positivecorresponds to the fact that the point qhas no volume, so not all Fi can be moved outwardsif they are all to meet at one point.Changing the origin, or equivalently translating ourconvex polyhedron, then modifies our effective divisor

e1F1 + ...+ euFu

by adding to it all possible principal divisors.

To say that we do not care about translatingour polyhedron is the same as saying thatwe don’t care about the divisor, only thedivisor class.

And we see that the divisor class groupis spanned by the Fi as an abeliangroup, with n linearly independentrelations coming from translation. Soit is an abelian group of rank u− n,the number of faces of S minusthe rank of the lattice which S spans.

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Now, I have cheated a bit because in defininglinearly equivalent divisors one is suposed touse principal divisors coming from arbitrarymeromorphic functions, here we only usedtorus invariant principal divisors. But, thatis OK because if a difference of torus invariantdivisors is principal it is certainly a principaltorus invariant divisor.

Example.

Consider the four elements of Z2 which are thecolumns of the matrix

(0 1 0 10 0 2 1

).

The slanted face connecting the lasttwo points has coefficient 2, the ones containing 0have coefficient 0, the remaining one has coefficientone. This is because the slantedface mets latice points if it is to betranslated to the origin.

An example of a principal divisor is the differenceof the two vertical edges minus the slanted edge.This is the divisor of the lattice point (1, 0) .

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Exercise. For convex sets spaning Zn showthat a torus equivariant map from one projective toricvariety to another amounts to saying that after expand-ing the first convex set by an integer multiple and takingthe convex hull again, it is a union of tanslatesof the other.

Since the convex set in the previous example already isa union of three vertical line segments, then there isa map from this complex projective surface to theprojective line (Rieman sphere).

That map is in fact a fiber bundle map,and our surface is a fiber bundle of the Riemannshpere over the Rieman sphere. It is theHirzebruch surface, also called a scroll.

There is a ring called the Chow ring which isa historical precursor to the cohomology(and for smooth projective toric varieties isthe commutative ring is the even dimensional part ofinteger cohomology).

Here we could define it by saying thatit is a graded ring which is Z in degree 0.In degree 1 it is spanned by the Fi modulothe n linear relations.

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Then we say that the ring multiplicationcomes from intersection, and the ring indegree i is spanned by codimension i faces.More precisely, if two faces f, g intersecttransversely the product fg is the intersection,and if they do not, relations in the divisor classgroup are used to arrange that they do.

Ampleness, Local Principality

We started with a finite set S containing an origin ofthe lattice, and interpreted the linear combinatione1F1 + ...+ euFu to mean that we have movedeach Fi outwards by ei steps so that the Fi

define the convex hull of S. Note that if we replaceeach ei by the same multiple λeiwhere λ is a positive integer, it will definethe convex hull of the equivalent set λS.

There is a definition that is not very relevant fortoric varieties, a divisor D is called ampleif some positive integer multiple λD isvery ample. In view of the statement above,

Observation A divisor on a toric variety isample if and only if it is very ample.

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Let’s give an explicit criterion of ampleness. Assumingthat two finite sets S, T both span the same latticeZn, the projective varieties defined by S and Tare the same if and only if some expansion λSof S by a positive integer λ is a union oftranslates of T and vice-versa. Thus when S is convex,a divisor a1F1 + ...+ auFu is ample (=very ample)if and only if for some positive integer λthe set of points enclosed in what we callwe call λa1F1 + ...+ λauFu is a unionof translates of S. Thus

Fact. A divisor D is (very) ample if andonly if the convex hull of the set of torus characterswith poles no worse than D is combinatorially equivalentto the convex hull of S, with corresponding faces of alldimensions parallel.

Let’s now give a description of the local definingequations of each divisor Fi as a codimension onesubset of our variety.

If we choose a vertex of S then a correspondingopen subset of the variety (it is covered by suchopen sets) consists of the affine maps S → Csending this vertex to 1 ∈ C. Those pointsbelonging to Fi are as we said just thoseaffine maps sending the points of S whichare enclosed in the polyhedron which we visualizeas e1F1 + ...+ (ei − 1)Fi + ...+ Fu to zero.

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If the chosen vertex does not belong to Fi thiswill be empty; while if the the chosen vertex doesbelong to Fi it will be those affine maps whichnot only send the chosen vertex to 1 but also sendthose points of S which do not belong to the faceFi. to zero. That is, if we view our chosen vertexas the origin of the lattice, we are talking about monoidhomomorphisms from the lattice span of S to C×

which send the monomial ideal spanned by the elementsof S which are not in Fi to zero.

To say that the divisor Fi is locally principal is tosay that for each choice of vertex (or equivalently forchoice of an element of S) this monomial ideal is locallyprincipal.

For each vertex, the ideal is generated bya single element if the divisor Fi

restricts to an actual principal divisor on the opensubset corresponding to each vertex. If S is convexand ei sufficiently large, this is the same assaying that the combinatorial type of S – that is,the combinatorial type of the polyhedron definedby the expression e1F1 + ...+ euFu – is thesame as the combinatorial type of the polyhedron definedby the expression e1F1 + ...+ (ei − 1)Fi + ...+ Fu.

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And that is equivalent to ampleness (and also then tovery ampleness) of that divisor.

Let’s write that down. Suppose that S is convex. Then

Fact: For each i, the following are equivalent

i) The defining ideal of the subvariety where Fi

meets the affine open subset corresponding to eachvertex of S is not only locally principal, but actuallyprincipal.

ii) for some (equivalently all) ample divisors H there isa positive integer λ such that λH − Fi is ample.

If the manifold is smooth then the monoid spannedby S when each vertex is viewed as the origin isjust a free commutative monoid on n generators,and every locally free ideal is free.

This shows that when the manifold is smoothwith initially chosen very ample divisore1F1 + ...+ euFu, after we multiplyall ei by the same number λto make them large enough then for each ithe divisor e1F1 + ...+ (ei − 1)Fi + ...+ Fu remainsvery ample.

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Arguing analagously for a divisor which is not justa single face,

Proposition. Let H be a an ample divisoron a smooth projective toric variety, and D anydivisor. Then there is a (positive) numberλ such that λH +D is ample.

For varieties which are not toric (even if they arestill assumed to be smooth) this is false.

In the special case when we take a1, ..., au tobe positive integers, and e1F1 + ...+ euFu

our original very ample divisor, as long asλ is large enough, what we see isthat the combinatorial type of theconvex hull of the set of of torus charac-ters with poles no worse than(λe1 − a1)F1 + ...+ (λeu − au)Fu is the same asthe the combinatorial type ofthe convex hull of the original set Sof torus characters with poles no worse thane1F1 + ...+ euFu. And the former is apolyhedron contained in the expansion ofthe latter by the number λ, resultingfrom expanding by λ, then movingthe face which we call Fi inwards byai steps in the cyclic quotient group Zu/Fi.

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When the variety is not smooth (when Scontains vertices not part of an affine latticebasis in the boundary) there can be a change ofof combinatorial type between one of theone of the polyhedra and the other. It correspondsto a blowing down if an integer expansion of thefirst polyhedron happens to be a union of latticetranslates of the second, and a blowing up if an integerexpansion of the second polyhedron happens to bea union of lattice translates of the first.

Note that the smaller polyhedron is not degeneratesince we chose the ei to be large compared to the ai.

The edges span the n− 1 degree termof the Chow ring, and the multiplication of thedivisor a1F1 + ...+ auFu with an edge Y is anelement of the n degree term, which is naturallyisommorphic to the ordinary integers, spannedby the class of any point of the variety.

The integer which corresponds to the producta1F1Y + ...+ auFuY is the the number of latticepoints (=torus characters) in Y as an edgeof the original polyhedron, minus the number oflattice points in the corresponding edge of thesmaller polyhedron.

That number can be positive or negative.

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Example.

The root system of type A2 (using the centerlessmodel of the group so that the torus is not artificiallyextended) is made starting with an affine basis ofthe two-dimensional lattice of characters, andmaking a hexagon from six such triangles usingreflections. Once we double the size of the hexagoneach edge contains three lattice points, whileif moved inwards one step contains four.Although the product of two consecutiveedges is 1, because they meet at one point(either thinking about them as edges of the hexagonor as projective lines in a variety they meet at thesame point), the self-intersection of theprojective line corresponding to each edge is3− 4 = −1. We also see from how we can move inthree disjoint edges two different ways to maketwo different triangles that we can blow down threenon-intersecting projective lines two different waysto obtain the projective plane. The resultingbirational transformation of the projective planeis called the quadratic transform.

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Example. Let’s return to the example where we startwith S the set of points of Z2 which are thecolumns of the matrix(

0 1 0 10 0 2 1

).

The convex hull includes also the point (0, 1).The four codimension-one faces are labelledA,B,C,D. Calling the complex manifold M, it isnonsingular since the triangle where A and B

meet is a lattice basis. It follows thatthe Chow ring is Z ⊕H2(M,Z) ⊕H4(M,Z), and let uscalculate it.

The commutative group H2(M,Z) is spannedby A,B,C,D with two relations coming fromtranslation. If the face B is moved outward (tothe left) while the face D is movedinward (to the right) the diagram iscongruent to the same diagram if theface A had been moved outward (upward)to the next coset in the lattice. Thusleftward translation in the lattice givesthe relation

B −D = A

in the Cohomlogy

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group. Upward translation gives

A = C.

The pairwise intersections of adjacent facesare four actual points of the manifold whichwe might label AB,BC,CD,DA. Becauseof the two relations we can replace A byC. Since A and C are disjoint thisimplies that in the Chow ring 0 = A2. Andwe can replace B by C +D. Then the fourpoints become C(C +D), (C +D)C,CD,DC.All of these are equal since C2 = 0. Weknew anyway that H4(M,Z) = Zand any point is Poincare dual to thefundamental class.

The elements C,D map to a basis of H2

and the product CD is a basis of H4.

We see from the diagram that D2 = −1because when the face D is moved inwardsit becomes one bit longer. Here the number −1denotes minus the fundamental class of M .This also can be derived from the relation D = B − Aas then D2 = DB −DA and the first term iszero while the second term is −1.

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Chapter 3 – Fourier series

Recall that I mentioned that if F1, ..., Fm arethe faces of a convex lattice polyhedron then anydivisor is equivalent to one of the type a1F1 + ...+ amFm

for a1, ..., am integers. If the ai are positiveso one may speak of the the polyhedron described-described by the symbol a1F1 + ...+ amFm, thenthe set of lattice points in the interior are thetorus characters with poles no worse thana1F1 + ...+ amFm

In fact, the vector space of all meromorphicfunctions on the projective variety with polesno worse than a1F1 + ...+ amFm is described bythis set of characters. That is, it is the finitedimensional vector space with torus action which isthe direct sum of a one-dimensional representation(with multiplicity one) with character each interiorlattice point.

A way of understanding this is to first note thatjust because the variety is projective, the vectorspace must be finite-dimensional (the finite-dimensionality can be taken to be a definitionof what it means for a variety to be projective,by the way, and the condition once verified forthe powers of any ample divisor holds for everydivisor).

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And then because the torus action preserves thefinite-dimensional space it decomposesinto one dimensional representations; and thesecan have no multiplicity larger than one as the wholefunction field has no component of multiplicity largerthan one.

The relation between the one dimensionalrepresentations and their characters is exactly theone which occurs in Fourier theory, and one couldcreate a projective toric variety by starting with aninfinite Fourier series in several variables, andputting bounds on the characters to describe a finitesubset. The convex hull of this set arises naturallythen, and is associated to a projective toric varietywith an ample divisor (polarization), and the space ofcorresponding Fourier series is the space of meromorphicfunctions with poles no worse than the chosen ample di-visor.

Such a bounding process is done in the classical way thatone obtains number-theoretic approximations of spectralfrequencies of the atoms, and there one also considersdifferential forms (as one can do of course for toricprojective varieties in general).

In that case the projective variety is not toric andso the restriction is not as simple as bounding an areain a lattice. Its geometry in any case is not relatedto the physical world; but is caused by choices which

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one makes in order to simplify the physical world; forexample choices of what are called ‘electron con-figurations.’

Yet, it is needed to know the geometry of one’s ownchoices before one can use coordinates to describe thenatural world. For, it is a projection of infinite seriesonto finite series which leads to being able to writecalculations in a finite amount of time; and typically theprojection requires an assumption of a Hilbert spacestructure or bilinear form. In general one can considerprojective geometry to be the ‘finitization’ of infiniteseries in analysis, and the issue is that one does notusually only want to know the exact value of a quantity,for instance at one time and one purported ‘point’ ofspace. Rather one wants to understand things more likeideas; as one contemplates a picture of subtle complexityit is not only that one wants to know, where exactly isthis ant orthat grain of dust, but more meaningful things.

The passage from an infinite calculationto a finite calculation must it seems be done based onintuition only, and the geometry of the choice which onehas made does reflect something which is internal to theobserver, only. In that sense, projective geometryconstitutes introspection rather than observation;and yet without introspection, observation of natureand then action amount to recklessness.

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