Alg. Geom. Regular function confusion

In summary, the lecturer discussed a map from a set in projective 2-space to projective 1-space, represented by \phi = (s_{0}:s_{1}) = (s_{2}:s_{0}). There was confusion about whether this was an alternative representation for the same function or a description for when the other is not defined. It was determined that the second option was correct because if s_{0} is equal to 0, then the other two values must also be 0, which is not a point in projective space.
  • #1
Bleys
74
0
So in class today the lecturer gave a regular map on the set [itex]V(s_{1}s_{2}-s_{0}^2)[/itex] in projective 2-space to projective 1-space by [itex]\phi = (s_{0}:s_{1})=(s_{2}:s_{0})[/itex].
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.
 
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  • #2
this makes no sense. your input point is not a point of P^2.
 
  • #3
?? [itex]\phi[/itex] takes a point [itex](s_{0}:s_{1}:s_{2})[/itex] to [itex](s_{0}:s_{1})[/itex] and/or [itex](s_{2}:s_{0})[/itex]
 

FAQ: Alg. Geom. Regular function confusion

What is algebraic geometry?

Algebraic geometry is a branch of mathematics that studies geometric objects defined by polynomial equations. It combines techniques from both algebra and geometry to understand the properties of these objects.

What is a regular function in algebraic geometry?

In algebraic geometry, a regular function is a function that is defined by a single polynomial equation and is well-behaved on a given algebraic variety. It is also known as a morphism, as it maps one algebraic variety to another.

What is the difference between an algebraic variety and a regular function?

An algebraic variety is a set of points defined by polynomial equations, while a regular function is a function defined on an algebraic variety. In other words, an algebraic variety is a geometric object, while a regular function is a mathematical function.

How do I determine if a function is regular?

To determine if a function is regular, you must first define the algebraic variety on which the function is defined. Then, you can check if the function is well-behaved on this variety by seeing if it can be defined by a single polynomial equation. If it can, then it is a regular function.

What are some applications of algebraic geometry?

Algebraic geometry has many applications in various fields, including physics, coding theory, and cryptography. It is also used in computer graphics to generate and manipulate geometric objects. Additionally, it has connections to other areas of mathematics, such as number theory and topology.

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