The dimensions of locus that is intersection of loci

[swampwiz]

It seems to me that for a set of loci of cardinality M having dimensions D_i in a space of dimension N, aside from degenerate intersections (e.g., a pair of spheres that touch at a single point), the dimension of the net intersection locus L is:

L = N - ∑ ( N - D_i ) = ( ∑ D_i ) - N ( M - 1 )

pair of lines in a plane: L = ( 1 + 1 ) - { 2 } ( { 2 } - 1 ) = 0 -> point

pair of planes in 3-D space: L = ( 2 + 2 ) - { 3 } ( { 2 } - 1 ) = 1 -> line

3 planes in 3-D space: L = ( 2 + 2 + 2 ) - { 3 } ( { 3 } - 1 ) = 0 -> point

plane & line in 3-D space: L = ( 2 + 1 ) - { 3 } ( { 2 } - 1 ) = 0 -> point

And for the case of linear loci, this can be proven in linear algebra. Then for the case of generally shaped loci, there is a topological correspondence between that loci and a linear loci of the same dimension (i.e., the linear loci can be stretched into whatever shape the general loci is).

Is this accurate?

 

[fresh_42]

It seems to me that for a set of loci of cardinality M having dimensions D_i in a space of dimension N, aside from degenerate intersections (e.g., a pair of spheres that touch at a single point), the dimension of the net intersection locus L is:

L = N - ∑ ( N - D_i ) = ( ∑ D_i ) - N ( M - 1 )

pair of lines in a plane: L = ( 1 + 1 ) - { 2 } ( { 2 } - 1 ) = 0 -> point

pair of planes in 3-D space: L = ( 2 + 2 ) - { 3 } ( { 2 } - 1 ) = 1 -> line

3 planes in 3-D space: L = ( 2 + 2 + 2 ) - { 3 } ( { 3 } - 1 ) = 0 -> point

plane & line in 3-D space: L = ( 2 + 1 ) - { 3 } ( { 2 } - 1 ) = 0 -> point

And for the case of linear loci, this can be proven in linear algebra. Then for the case of generally shaped loci, there is a topological correspondence between that loci and a linear loci of the same dimension (i.e., the linear loci can be stretched into whatever shape the general loci is).

Is this accurate?

Not really. The obvious flaw is, that you disregarded parallel or skew objects. The next point is, that the terms dimension as well as cardinality aren't well-defined the way you use them, esp. in the non-linear case. It is quite a bit of work necessary to make them mathematically useful. As an example for your naive use of the word topological, consider this example:
https://en.wikipedia.org/wiki/Hilbert_curve
How does a Hilbert curve fit in your framework?

 

[WWGD]

There is a definition for objects in "General Position". The dimension of the intersection is the difference of the sum of dimensions of intersecting objects subtracted from dimension of ambient space. So, e.g., in $\mathbb R^2$ , two lines in general position have an intersection of dimension 2-(1+1)=0, so they intersect at a point.This formula applies for manifolds in Euclidean space.

 

[swampwiz]

Not really. The obvious flaw is, that you disregarded parallel or skew objects. The next point is, that the terms dimension as well as cardinality aren't well-defined the way you use them, esp. in the non-linear case. It is quite a bit of work necessary to make them mathematically useful. As an example for your naive use of the word topological, consider this example:
https://en.wikipedia.org/wiki/Hilbert_curve
How does a Hilbert curve fit in your framework?

I consider parallel or skew objects to be degenerate. And I consider the Hilbert curve (I had no idea this exists!) to be a contrived example, like one of Weierstrauss's non-continuous functions. OK, I understand that this is not "rigorous".

 

[WWGD]

And still, for general objects you can intersect homology classes.You also can use tangent planes here.

 

[swampwiz]

There is a definition for objects in "General Position". The dimension of the intersection is the difference of the sum of dimensions of intersecting objects subtracted from dimension of ambient space. So, e.g., in $\mathbb R^2$ , two lines in general position have an intersection of dimension 2-(1+1)=0, so they intersect at a point.This formula applies for manifolds in Euclidean space.

OK, so it sounds like I am correct. Obviously, the case for linear systems is straightforward. How is this proven for general systems?

 

[WWGD]

OK, so it sounds like I am correct. Obviously, the case for linear systems is straightforward. How is this proven for general systems?

I think you linearize locally, using tangent spaces for the case of manifolds. IIRC you can use homology classes, as their intersection is a well-defined homology class.

 

[mathwonk]

intersection theory is a very interesting and important topic, and your intuition is substantially correct about what happens except in "degenerate cases". the trick is to make more precise what one means by degenerate cases and say something even for those. in algebraic geometry we remove some degenerate cases by restricting to sets defined by polynomials, (hence excluding weierstrass curves), and also consider only algebraically closed fields, (hence excluding spheres that meet at a single point), and we also prefer to work in projective space, (hence excluding parallel phenomena). The result is that any two irreducible algebraic sets of dimensions n, m in projective spave of dimension r, such that n+m ≥ r, must in fact always meet in a non empty set, and moreover every irreducible component of that intersection has dimension ≥ n+m-r.

for this result, see Hartshore's Algebraic Geometry, chapter 1, theorems 7.1. and 7.2, p. 48.

the more advanced and subtle aspect of algebraic intersection theory is to assign degrees to the intersection compomnents and make statemens about the degree of the intersecvtion in terms of the degrees of the sets being intersected. E.g. in the projective plane, two distinct irreducible curves of degrees d,e meet in at most de points. A good introduction to the subject is the book Algebraic curves by Robert Walker, and with more algebraic background, the book of the same name by William Fulton, available free on his website
http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf
. The definitive modern treatment is the very advanced book Intersection theory by Fulton.

 

[Infrared]

In a manifold $M$ of dimension $n$, if $X,Y$ are submanifolds of dimensions $r,s$ that intersect transversely (this rules out things like tangent spheres as you've noted), then $X\cap Y$ is a submanifold of dimension $n-(n-r)-(n-s).$

This can be trivially restated in following way that maybe makes the result a bit easier to remember: define codimension of a submanifold $N\subset M$ to be $\dim(M)-\dim(N)$. Then in the above situation, you have $\text{codim}(X\cap Y)=\text{codim}(X)+\text{codim}(Y).$

 

[mathwonk]

yes, the diferentiable theory of (transverse) intersections, and the approximation of arbitrary ones by transverse ones, is nicely treated in the book by Guillemin and Pollack.

 

[Infrared]

Certainly Guillemin and Pollack is a great text (and the one I learned from). To the OP, a proof of the theorem I stated above can be found on page 28.