Algebraic varieties in a minimal space

disregardthat

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Are there any methods for finding the minimal (or at least of smaller dimension) projective space P^n which can contain a given algebraic variety?

For example, some curves in P^2 are isomorphic to P^1, and so P^1 is the minimal space which can contain such a curve.

I am looking for something like an inverse segre-embedding (in some sense) or an inverse d-uple-embedding, only more general.

Any clues?
 
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mathwonk

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every non singular algebraic variety of dimension n can be embedded in P^(2n-1) I believe. [This is wrong, see below.] Let's see from the usual argument: first embed it somewhere, then try to project down. the projection will be an embedding iff the point from which we project does not lie on any secant or any tangent, so we need to estimate when we can avoid all secants and tangents.

the union of the set of all secants has dimension ≤ 2n+1, so as long as we are in a higher dimension than that we can avoid them. tangents are secants at the same point so their union has dimension ≤ n+1, so that is ok too.

so i was wrong, it is not 2n-1, but 2n+1. and this says e.g. all smooth curves embed in P^3.

since the union of all tangents has dim n+1 we can immerse a smooth variety in P^(n+1), so all smooth curves immerse in the plane.

the more refined question of which curves embed in the plane and which surfaces embed in P^3 or P^4 is also interesting.
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If singularities are allowed there seems to be no bound to the embedding dimension.

the only curves in P^2 which are isomorphic to P^1 are lines and smooth conics.

The most fundamental invariant of a curve is its topological genus. most, but not all, smooth curves of genus 3 embed in the plane. however smooth curves of genus 2 never embed in the plane. all smooth curves of genus 1 embed in the plane.

a smooth curve that embeds in the plane always has genus equal to a triangular number, i.e. an integer of form (1/2)(n)(n-1). e.g. if n = 3, we get g = 1, and if n = 4 we get g = 3. but we never get g = 2, or 4 or 5. n=5 gives g = 6....... (n is the degree of the plane curve.)


hartshorne's algebraic geometry book discusses the restriction placed on a smooth surface by the riemann roch theorem, to be embedded in P^4 or P^3 I believe. (all smooth surfaces embed in P^5.)
 
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disregardthat

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every non singular algebraic variety of dimension n can be embedded in P^(2n-1) I believe. [This is wrong, see below.] Let's see from the usual argument: first embed it somewhere, then try to project down. the projection will be an embedding iff the point from which we project does not lie on any secant or any tangent, so we need to estimate when we can avoid all secants and tangents.

the union of the set of all secants has dimension ≤ 2n+1, so as long as we are in a higher dimension than that we can avoid them. tangents are secants at the same point so their union has dimension ≤ n+1, so that is ok too.

so i was wrong, it is not 2n-1, but 2n+1. and this says e.g. all smooth curves embed in P^3.

since the union of all tangents has dim n+1 we can immerse a smooth variety in P^(n+1), so all smooth curves immerse in the plane.
Thanks, I will look into the details of this. But I have a question. Why is the condition that the point from where we project must not lie on a secant or tangent sufficient? Where are we projecting to?
 

mathwonk

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suppose you project a space curve from a point p in P^3, to the plane. Each point q on the curve goes to the point of intersection iof the plane with the line [pq].

Thus if p lies on the secant qr, then both lines [pq] and [pr] meet the plane at the same point and the projection is not in jective, sending q and r to the same point.

If p lies on the tangent line of the curve at q, then the projection collapses the tangent line [pq] to zero, hence the projection is not injective on the tangent space to the curve at q, hence is not a local isomorphism there.

details can be found in shafarevich chapter II.5.5, or (for curves) Hartshorne pp.309-310, or (form curves) Mumford AG I, pp.132-136, or (for diff manifolds) Guillemin - Pollack, chapter I.8, in particular p.51, or Dieudonne' Foundatiopns of modern analysis, vol. III. chapter XVI.25, and problems 2;13c, or Auslander and Mackenzie, chapter 6., etc.......
 

disregardthat

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I read 309-310 in hartshorne and I understand you better now.

I have a projective variety of dimension 2 in a large projective space (with many generators for the ideal), but I suspect that it can be contained in P^3. Considering that almost all generators are of degree 2, it is likely contained in a d-uple embedding. How would you suggest I go about finding a smaller space for my variety?

I had a curve in P^5, and one could regonize from the generators that the curve was contained in a 2-uple embedding of P^2, so I was able to pull the curve back to P^2 using such an embedding. I am looking to do something similar for the other variety.

Do you think it is computationally feasible to go from, say, P^30 to P^3 if it was possible for my variety?
 

Bacle2

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What is the main advantage of having varieties embedded in projective space? Is it the intersection results, i.e., controlling the way that the curves/surfaces intersect?
 

mathwonk

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yes. It tells you more about their geometry and function theory.

E.g. a curve of degree 3 in the plane has a function with two poles, because you can project from a point of the curve to a projective line.

Equivalently the inverse images of specific points of the line under the function, are the additional intersections of the curve with lines through the point of projection.


It also tells you something about the topology of the variety. E.g. all smooth surfaces in complex projective 3 space are simply connected, by Lefschetz' theorem.

You can also study the geometry by intersection theory. E.g. there is a 19 dimensional family of cubic surfaces in P^3 and a 4 dimensional family of lines. But it is also 4 conditions for a line to lie on a cubic surface, so every line lies on a P^15 of cubics in the full P^19 of cubics.

Hence one expects every cubic surface to contain a finite number of lines, and indeed each smooth surface contains 27 of them.

One can also study varieties in projective space by letting them vary in families, by varying their equations. This is analogous to using induction.

By fixing one line on a cubic surface and looking at all planes through that line, one fibers the cubic surface into a pencil of conic curves, another useful representation of the surface.
 
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