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I've been studying Algebraic Geometry (in coding theory), and my book has been very ambiguous about what some of these ideas actually look like. So I was curious as to what some of the Divisors of degree 2, degree 3, and degree n Fermat Curves are.

KaneZ
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  • A Fermat curve is a curve in $\mathbb{P}^2$ given by $X^n+Y^n=Z^n$ for some $n$. A divisor on it is a sum of points with coefficients. What do you want to know more on it? – Jérémy Blanc Apr 01 '14 at 15:19
  • I would like to see some concrete examples of divisors, for degree 2 or 3 so I can see degree n a little easier. Since it is for coding theory, calculating them and knowing concrete examples is important. – KaneZ Apr 01 '14 at 15:21
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    Divisors of degree $d$: choose for example $d$ points on the curve and add them. You can also take one point $(-1)$ times, and add $d+2$ other points. The degree of the divisor is the sum of the coefficients associated to each point. It has nothing to say with the fact that the curve is Fermat. – Jérémy Blanc Apr 01 '14 at 15:24
  • @Jérémy's answer is otherwise ok, but ignores the possibility that some points may have their coordinates in an extension field of the field of definition. To make the usual formulas work you need to be careful when tallying the points, and take the degree of the field extension generated by their coordinates into account. Furthermore, points with conjugate coordinates count as only a single place. So if you look at the Fermat curve $x^3+y^3+z^3=0$ over the field $\Bbb{F}_2$, it has a degree one point $P=(1,1,0)$ and a degree two point $Q=(1:z:0)$, where $z$ is an unspecified cubic root – Jyrki Lahtonen Apr 05 '14 at 08:46
  • (cont'd) of unity that only exists in the extension field $\Bbb{F}_4$. Thus the divisor $P+Q$ has degree 3. When you extend the field of constants, then the point $Q$ becomes a pair of conjugate points $Q_1=(1:A:0)$ and $Q_2=(1:B:0)$, where $A$ and $B=A^2=A+1$ are the primitive third roots of unity in $\Bbb{F}_4$. Leaving these as a comment as I cheated a bit and didn't explain the difference between a point and a place. In general (and this comes to the fore in coding theoretical applications) a divisor is an integer linear combination of places, and conjugate points form a single place. – Jyrki Lahtonen Apr 05 '14 at 08:51
  • I was in fact talkin about a divisor over an algebraically closed field. Then you look at divisor defined over the subfield (invariant by the Galois action) and you get the divisors. But of course you could avoid this and try to stay on the subfield, but this complicates the definition. – Jérémy Blanc Apr 05 '14 at 14:42
  • @JérémyBlanc: Guessed that much. To me here the context (coding theory) very strongly suggests that the ground field is finite, so that extra problem is unavoidable. – Jyrki Lahtonen Apr 06 '14 at 12:02

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