Alg. Geom. Regular function confusion

Click For Summary
SUMMARY

The discussion centers on the mapping of a regular function defined on the set V(s_{1}s_{2}-s_{0}^2) in projective 2-space to projective 1-space via the function φ = (s_{0}:s_{1})=(s_{2}:s_{0}). The confusion arises regarding whether this mapping represents an alternate description of the same function or indicates a different behavior where the function is not defined. It is established that if s_{0}=0, then s_{1} and s_{2} must also be zero, which does not correspond to a valid point in projective space. Therefore, the conclusion is that this mapping is indeed an alternate description where the original function is not defined.

PREREQUISITES
  • Understanding of projective geometry concepts, specifically projective spaces.
  • Familiarity with regular maps and their properties in algebraic geometry.
  • Knowledge of the implications of points at infinity in projective spaces.
  • Basic grasp of function representation in mathematical contexts.
NEXT STEPS
  • Study the properties of regular maps in algebraic geometry.
  • Explore the concept of points at infinity in projective geometry.
  • Learn about the implications of defining functions in projective spaces.
  • Investigate examples of mappings between projective spaces and their geometric interpretations.
USEFUL FOR

Students and educators in mathematics, particularly those studying algebraic geometry and projective geometry, will benefit from this discussion. It is also relevant for anyone seeking to clarify the behavior of functions in projective spaces.

Bleys
Messages
74
Reaction score
0
So in class today the lecturer gave a regular map on the set V(s_{1}s_{2}-s_{0}^2) in projective 2-space to projective 1-space by \phi = (s_{0}:s_{1})=(s_{2}:s_{0}).
I'm confused. Is that another representation of the "function"? (Meaning they map to the same point classes?) or is it an alternate description on where the other is non defined?
I mean, it can't be the first option, since if the s_{0}=0 then the other two must be zero, but there is no such point in projective space.
But, I just want to make sure it really is the other option and not be something else I missed.
 
Physics news on Phys.org
this makes no sense. your input point is not a point of P^2.
 
?? \phi takes a point (s_{0}:s_{1}:s_{2}) to (s_{0}:s_{1}) and/or (s_{2}:s_{0})
 

Similar threads

Finding Eigenvalues of an Operator with Infinite Basis
  • · Replies 1 ·
Replies
1
Views
2K
Graduate How to calculate the Wigner function for an entangled state
  • · Replies 4 ·
Replies
4
Views
2K
MHB Checking if $f_1, f_2, f_3 Belong to $S_{X,3}$
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Fourier Transform of an exponential function with sine modulation
  • · Replies 5 ·
Replies
5
Views
2K
MHB Is f in the vector space of cubic spline functions?
  • · Replies 1 ·
Replies
1
Views
2K
Graduate What Determines the Degree of the Gauss Map in Differential Geometry?
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Attempting to find an intuitive proof of the substitution formula
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad How Do Transition Functions Define a Möbius Strip in Fiber Bundles?
  • · Replies 10 ·
Replies
10
Views
4K
Graduate Understanding the optimal approximate qubit cloning method
  • · Replies 0 ·
Replies
0
Views
1K
Graduate Smooth Homotopy, Regular Values (Milnor)
  • · Replies 4 ·
Replies
4
Views
3K