Proj

For the complex topology, any connected component of an affine variety is either a single point or non-compact. Indeed, if is a connected component of an affine variety, the functions can't have a maximum on unless they are constant so if is compact then each is equal to some constant on , i.e. is the point . This non-compactness of affine varieties is a motivation to introduce projective spaces and projective varieties.

Let be the space of complex lines in . We denote the points of . As a line is determined by some direction vector, well-defined up to scaling by non-zero scalars, can be identified with the quotient

Making use of the natural Hermitian structue on , i.eof the norm , we see that a line is well-defined by a unitary vector up to scaling by some scalar of unit norm. This gives an identification of with the quotient where denotes the -dimensional sphere. In particular, is compact for the complex topology. The space
is called the "projective space of dimension ".

In order to describe the projective space in terms of algebra, it is enough to work on in a -equivariant way. More precisely, we consider the action of on with acting on a degree homogeneous polynomial by multiplication by . Let us denote and the finite dimensional vector space of degree homogeneous polynomials. As any polynomial can be uniquely decomposed in homogeneous pieces, we have the direct sum decomposition

This decomposition is the eigenspace decomposition for the -action on : is the eigenspace of the action of for the eigenvalue . The space is called the "weight space of weight " of the -action on .

"'Exercise"': We say that a finite dimensional representation of is "algebraic" if the corresponding group homomorphism is algebraic. Show that the data of a -graded vector space with finite-dimensional graded components is equivalent to the data of a vector space with a -action such that every is contained in a finite-dimensional -invariant subspace on which the -action is algebraic.

Let be a degree homogeneous polynomial. Then, for every , we have if and only if . So it makes sense to say that vanishes at some point . An ideal of is called "homogeneous" if or equivalently if is generated by homogeneous polynomials. It makes sense to say that all the functions in a given homogeneous ideal of vanish at some point and so every homogeneous ideal of defines a subvariety of . In other words, the subvarieties of defined by the homogeneous ideals of are precisely the -invariant subvarieties of and so define subvarieties of . The -invariant subvariety of associated to a subvariety of is called the "affine cone" of this subvariety. A subvariety of is called a "projective variety".

Let be a homogeneous ideal of . Then the quotient ring is also graded:

We define the "homogeneous spectrum" of the graded ring , denoted , as being the set of homogeneous ideals of which are maximal amongst homogeneous ideals, except the so called "irrelevant ideal", which is the ideal of the origin in the -invariant subvariety of defined by , and thus corresponds to the empty set after passing to projective space. It is possible to show that can be naturally identified with the set of points of the projective variety defined by . In other words, if are homogeneous polynomials in , we have

"'Example"': Let be the projective curve in defined by the degree 3 homogeneous polynomial . Its affine cone (with real coordinates) is a union of lines through the origin. Consider its intersection with the plane . We can identify with the plane affine curve via the map . We think of the remaining points on as "at infinity" with respect to this affine set.


"'Exercise"': Rewrite the table of dualities by replacing by , ideals by homogeneous ideals, rings by graded rings, by ...

"'Exercise"': Work out the details of . Pass back and forth between homogeneous polynomials in and polynomials in .

A degree homogeneous polynomial is a function on but is not a function on . A natural question is: what is the interpretation of the degree homogeneous polynomials in terms of the projective space ? For , a degree homogeneous polynomial defines a function on the line which is of degree . For example, if , the coordinates are linear functionals on the line . We call the line and the union of these lines, i.e.

As the fibres of the natural map from to , , are copies of , i.e. lines, we see that is a "line bundle" on . As the fibre over a point of , i.e. over a line in , is precisely this line, is called the "tautological" line bundle on .

Let us consider the natural map from to , . The fibre of over is the set of lines passing through . If is non-zero, there is a unique such line and so is one-to-one above . If , any line passes through zero and so the fibre is . This map is called the "blow-up of the origin in ": one goes from to by separating the various lines intersecting each other at the origin and this effectively replaces the origin of by the projective space .

As the coordinates are linear functionals on the lines , the natural interpretation of the 's is as sections of the dual line bundle of , called . More generally, degree homogeneous polynomials naturally define sections of the line bundle obtained by the -th tensor product of the dual of , i.e. of . This line bundle is called .

"'Exercise"': What is the interpretation of the quotient of an affine variety by the action of a finite group in terms of coordinate rings?

"'Exercise"': Prove the degree-genus formula, a smooth degree curve in is of genus , by degenerating the degree curve to a union of lines.

"'Exercise"': Let be a "reasonable" topological space. Show that the closed subsets of are exactly the subsets of the form where the are continuous functions on . Now invent the Zariski topology by replacing continuous functions by polynomials.

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