The quot functor and the hilbert polynomial strati cation. Sending x to the set of rank n vector bundles together with a subbundle of rank d. More generally, hypersurfaces of degree din pnare naturally parameterized by the projective space associated to h0pn. Most families of curves in projective space specialize to singular curves. Lines in projective space mathematics stack exchange. Vector bundles on projective space takumi murayama december 1, 20 1 preliminaries on vector bundles let xbe a quasi projective variety over k. Chapter 5 homogeneous representations of points, lines. Naturally, vector spaces over c are not alone in this respect. The functor which sends r r to the r rpoints of the projective space. Below we will give a few details explaining how both constructions work.
In fact, the converse holds for a suitable class of functors. Projective varieties are zeroes of systems of homogeneous polynomial equations in projective n ndimensional space. We will now investigate these additional points in detail. Projective geometry for perfectoid spaces gabriel dorfsmanhopkins university of washington june 21, 2018. These are the projective homogeneous spaces with reduced stabilizer group schemes.
Grp is representable by a closed subscheme of gl n2. There is another serious problem with the functor mg. For a finite dimensional real vector space v with inner product, let fv be the block structure space, in the sense of surgery theory, of the projective space of. Yes, a line in the projective space associated to a vector space is a plane in that vector space. Projective space article about projective space by the free. Both methods have their importance, but thesecond is more natural. The tangent space of this functor is given by grothendiecks 7 equivariant cohomology group h1x,g,tx, 1, 3. A functor from schemes to sets that arises this way is called representable. If v is a complex analytic subvariety of pure dimension k in the unit ball in c which does not contain the origin, then the 2fcvolume of v equals the measure computed with multiplicity of the set of n a. We introduce the cech cohomology and use it to calculate cohomology. Another example of a projective plane can be constructed as follows.
As we know, the projective space pn represent the functor sending x to the set of line bundles l on x together with a surjection from the trivial vector bundle to l. Projective geometry is designed to deal with points at in. Grp be a presheaf, and recall that all limits and colimits exist in grp. A homogeneous space is a gspace xsuch that the induced action on kpoints gk. More relevant to our current discussion is the functor of points of projective space. The key ingredient for our definition of the functor of points of the relative scheme pn. Let be such a variety, so that is canonically embedded in the projective space for some sufficiently large. Complex projective space the complex projective space cpn is the most important compact complex manifold. Then, we say that f is stalky if, for any a ne speca and any p 2speca, the natural. Using the internal language of toposes in algebraic geometry, talk at toposes at ihes, november 2015 pdf, recording. Nevertheless, we are able to calculate this socle in some cases, using the socle of the burnside functor. The functor which sends r r to the r r points of the projective space. A projective frame is a set of points in a projective space that allows defining coordinates.
Gk,n naturally carries the structure of a smooth, projective variety. Hilbert schemes of points in projective spaces ut math. The hilbert scheme parameterizes subschemes of projective space with a xed hilbert polynomial, thus provides the starting point for all such constructions. C set is a zariski sheaf if for any object x with open. A basis for investigations on incidence projective spaces. A classical approach to investigate a closed projective scheme w consists of considering a general hyperplane section of w, because many properties of w are preserved under general hyperplane sections and can be easier recognized in subschemes of lower dimension.
The functor of points and the hilbert scheme clearly a scheme contains much more information than the topology of the underlying set. These notes are collected from talks given by the authors at the university of nice octoberdece 2002. Subsequently, i will recall the construction of the grassmanian and describe the moduli functor that it. Algebraic geometry starts with study of spaces that are locally. Projective space, proof of existence as a scheme, the line bundles, coor. This idea can be generalized and made more precise as follows. The points in the boundary correspond to npointed stable rooted trees of d. With every projective space defined in terms of collinearity relation we associate the incidence structure consisting of points and lines of the given space. We introduce the general notion of projective space defined in terms of incidence. Similarly, plane conic sections are parameterized by the 5dimensional projective space of ternary quadratic forms. For instance, the functor f f describing projective n nspace is given on fields by. Before we begin, we require some preliminaries about zariski sheaves. Yonedas lemma says that a scheme x determines and is determined by its points.
An elliptic curve gives an example of a projective group scheme, ie. More precisely, if two functor of points h x and h. The hilbert scheme provides a generalization of these examples to parameter spaces for arbitrary. For instance, the functor f f describing projective n n space is given on fields by. Nevertheless it is possible to consider a scheme as a hierarchy of sets of points. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. It is certainly fundamental, so the answer is likely contained at least implicitly in most sources out there, but i havent seen it done this way that is, in this particular functorial manner in a way which is overt enough for me to catch on. The main textbook for this course is qing lius algebraic geometry and arithmetic curves, 2006 paperback edition. We have to show that f is a morphism of projective spaces and that f in. Standard examples of projective homogeneous gspaces are the. Projective invariants of projective structures and applications by david mumford the basic problem that i wish to discuss is this.
The projective space pn thus contains more points than the a. Elementary surprises in projective geometry richard evan schwartz and serge tabachnikovy the classical theorems in projective geometry involve constructions based on points and straight lines. The reason we do not list this here is that we have not yet defined the notion of finite type at this point. My question is, what functor does the grassmannian grd,n represent. The set of one dimensional subspaces of v corresponds bijectively to the points of the projective space pv. Master mosig introduction to projective geometry a b c a b c r r r figure 2. Typically, only fieldvalued points of a scheme are easy to describe. Tqfts as homotopy xed points, and we show how anomalous ndimensional tqfts in codimension 1 give rise to projective representations of the mapping class group of closed n 1dimensional manifolds, hence to projective modular functors. A major problem in the study of smooth curves embedded in projective space is the relationship between the degree and genus of such curves.
In mathematics, a space is a set sometimes called a universe with some added structure while modern mathematics uses many types of spaces, such as euclidean spaces, linear spaces, topological spaces, hilbert spaces, or probability spaces, it does not define the notion of space itself. Leave a comment posted by dan bragg on november 20, 2015. More precisely, if two functor of points h x and h y are isomorphic as functors, then xand y are isomorphic as schemes. A family of vector spaces over xis a morphism of varieties e. We have and will continue to abuse notation throughout by identifying a representable scheme with its functor of. Let k be an infinite field, a be a noetherian kalgebra and p a n the ndimensional projective space over a. Jun 08, 2017 precisely, it is the functor of points of an a. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Algebraic geometry is, in origin, a geometric study of solutions of systems of polynomial equations and generalizations the set of zeros of a set of polynomial equations in finitely many variables over a field is called an affine variety and it is equipped with a particular topology called zariski topology, whose closed sets are subvarieties. This way of looking at schemes can be very pro table. Weibel received 25 may 1988 revised 28 june 1989 in this paper we study the graded minimal free resolution of a finite set of points in ip. As an application of general techniques of cohomology we prove the grothendieck and serre vanishing theorems. We introduce a smooth projective variety t d,n which compacti. In this construction, each point of the real projective plane is the onedimensional subspace a geometric line through the origin in a 3dimensional vector space, and a line in the projective plane arises from a geometric plane through the origin in the 3 space.
When we use this point of view, we will denote the grassmannian by gk. Yonedas lemma, the functor of points h x determines the scheme x. Functors of liftings of projective schemes sciencedirect. So it is nothing else than a chamber in the building of any thick projective plane. A similar phenomenon in algebraic geometry is given by a linear system. It is certainly fundamental, so the answer is likely contained at least implicitly in most sources out there, but i havent seen it done this way that is, in. In contrast to usual algebraic geometry, the functor spec. Pdf pointed trees of projective spaces researchgate. For example, a map to is the same as a globallydefined regular function. If you are funked out by this idea of understanding a space by thinking of its functor of points, think that this is a generalization of what we do when we talk about manifolds. The quot functor and the hilbert polynomial strati cation 5 6 10. Alternatively, we can ask for a delignemumford stack that parameterizes smooth curves. The original reference for this material is ega iii, but most graduate students would prob. The classes can be interpreted as the points of projective space, in which case the numbers in the quadruples are called the homogeneous coordinates of the points.
The rising sea foundations of algebraic geometry math216. The projective space associated to r3 is called the projective plane p2. This means concretely that gr i x i where x i is a subfunctor of gr represented by a projective scheme, such that x i. The functor gr is representable by an indscheme, which is ind projective over k. Betti numbers of points in projective space anna lorenzini dipartimento di matematica, via vanvitelli, i, 06100 perugia, italy communicated by c. Points in projective space in very uniform position abstract. We say that sis in linear general position if any subset of k n points spana a k 1plane. The hilbert scheme is a very di cult object to construct and it is normally done by showing it is a speci c case of a more general object known as the quot scheme. The functor of points and the hilbert scheme clearly a scheme. Projective space is one of the fundamental objects studied in algebraic geometry. Note that projective planes precisely are generalized.
To do so, i will show that the quot functor is a closed subfunctor of a relative grassmanian over s. It is often convenient to think of gk,n as the parameter space of k. Vector bundles on projective space university of michigan. The block structure spaces of real projective spaces and. One general way to study these invariants of curves was extensively exploited by g. The socle of a projective mackey functor for a pgroup.
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