What's the most basic yet interesting Algebraic Geometry result regarding this polynomial? Let $f(x,y,z) = x^a + y^b - z^c$, where $a,b,c \gt 0$.  What is the most basic yet interesting result about this polynomial from Algebraic Geometry?
 A: You might be interested in the paper Faltings plus epsilon, Wiles plus epsilon, and the Generalized Fermat Equation  by H. Darmon, which contains a number of interesting results and conjectures about the integer solutions to the equation $f(x,y,z) =0$. For instance, Darmon and Granville conjecture that there are finitely many (primitive) triples of integers $(x,y,z)$ with $x^a+y^b = z^c$ and such that $$1/a+1/b+1/c<1.$$ Moreover, Darmon offers a bounty of
$$300\left(\frac{1}{\frac1a+\frac 1b+\frac1c}-1\right)\:\text{dollars}$$
for any solution not in the list contained in the paper. 
(However, please don't search too hard, because Prof. Darmon is my doctoral advisor and I wouldn't want to cause his ruin. Of course, I am kidding: good luck finding any new solutions...)
A: In a similar vein as Bruno's answer, I invite you to take a look at this paper of Poonen, Schaefer, and Stoll, in which they find all the solutions to $x^2+y^3=z^7$ listed in "Faltings plus epsilon" using algebraic geometry. They also give a nice section summarizing why this case is especially hard and past results on the generalized Fermat equation.
