bitsmath said:Hello!
I would like to learn finding genus of my curve x^2-x+y-y^5 = 0. This is no doubt, it is hyper-elliptic curve and has genus 2. But, I do not know the technique of finding genus. Awaiting suitable reply.
Thanks in advance! Please explain
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mathbalarka said:Where did you find this problem?
For the question, Falting's theorem states that if the genus of a given curve is $> 1$ then it has finitely many $\Bbb Q$-points. So having the genus automatically answers your second question about Falting, so that part is superfluous.
Not really sure how to compute genus of this particular curve. Have you looked at the corresponding Riemannsurface over $\Bbb P^1$? Perhaps Riemann-Hurwitz or some such?[Sir, I got your message and I re-edited my question by equating to 0. Please explain how to find genus of my curve. This problem is self made.]
This is no doubt, it is hyper-elliptic curve and has genus 2.
mathbalarka said:Well, I have already given you a plausible way to approach this. There are pretty well-known formulas for calculating genus of Riemann surfaces, try them out! If you don't know how to compute genus of Riemann surfaces, try getting a decent book.
If you already know that it has genus 2, what is your question?
The genus of a curve is a mathematical term that refers to the number of holes or handles that a curve has. It is a topological property that helps classify and differentiate between different types of curves.
To determine the genus of a curve, you can use the Euler characteristic formula: χ = V - E + F, where χ is the Euler characteristic, V is the number of vertices, E is the number of edges, and F is the number of faces. The genus can then be calculated using the formula: g = (2 - χ)/2.
No, a curve cannot have a negative genus. The genus of a curve is always a non-negative integer, as it represents the number of holes or handles in the curve.
Yes, there are other methods to find the genus of a curve, such as using the Gauss-Bonnet theorem or the Riemann-Roch theorem. These methods involve more advanced mathematical concepts and are typically used for more complex curves.
No, the genus of a curve is not always unique. Some curves may have the same genus, while others may not have a well-defined genus. It is important to consider the specific properties and characteristics of a curve when determining its genus.