Lesson plan
Quartic surface
In mathematics, especially in algebraic geometry, a quartic surface is a surface defined by an equation of degree 4. More specifically there are two closely related types of quartic surface: affine and projective. An affine quartic surface is the solution set of an equation of the form : f(x,y,z)=0\ where is a polynomial of degree 4, such as . This is a surface in affine space . On the other hand, a projective quartic surface is a surface in projective space of the same form, but now is a homogeneous polynomial of 4 variables of degree 4, so for example . If the base field is or the surface is said to be real or complex respectively. One must be careful to distinguish between algebraic Riemann surfaces, which are in fact quartic curves over , and quartic surfaces over . For instance, the Klein quartic is a real surface given as a quartic curve over . If on the other hand the base field is finite, then it is said to be an arithmetic quartic surface.
Article summary
In mathematics, especially in algebraic geometry, a quartic surface is a surface defined by an equation of degree 4. More specifically there are two closely related types of quartic surface: affine and projective. An affine quartic surface is the solution set of an equation of the form : f(x,y,z)=0\ where is a polynomial of degree 4, such as . This is a surface in affine space . On the other hand, a projective quartic surface is a surface in projective space of the same form, but now is a homogeneous polynomial of 4 variables of degree 4, so for example . If the base field is or the surface is said to be real or complex respectively. One must be careful to distinguish between algebraic Riemann surfaces, which are in fact quartic curves over , and quartic surfaces over . For instance, the Klein quartic is a real surface given as a quartic curve over . If on the other hand the base field is finite, then it is said to be an arithmetic quartic surface.
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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