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edited Aug 26, 2012 at 14:36

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For question 2, you can look for surfaces of Picard number 1. One way to find such surfaces is to reduce modulo p and show that the reduction has Picard number 1 by computing its L-function. This will only work for odd degree. For even degree, you may need to compute the reduction modulo two primes with Picard number 2 but incompatible lattices so the Picard number of the original surface is 1 (this is a trick due to R. van Luijk). These computations are not easy and there is some work, e.g. by Kedlaya, to make them feasible.

Question 3 is false. You can easily write down smooth surfaces of any degree containing a straight line.

Off the top of my head, I don't have an example for your question 1, which is unconditional. But, if believe the ABC conjecture in its multiple summands variant, then diagonal hypersurfaces give examples (e.g. $x^n+2y^n+3z^n=6w^n, n$ large).

Edit: As Jason points out in the comments, my suggestion for question 2 is not enough. A different idea would be to use the idea for question 1. Namely use the function field ABC (which is a theorem) to show that a "fewnomial" (diagonal equations contain lines) equation has no curves of genus 0 or 1. I haven't worked out the details, though.

For question 2, you can look for surfaces of Picard number 1. One way to find such surfaces is to reduce modulo p and show that the reduction has Picard number 1 by computing its L-function. This will only work for odd degree. For even degree, you may need to compute the reduction modulo two primes with Picard number 2 but incompatible lattices so the Picard number of the original surface is 1 (this is a trick due to R. van Luijk). These computations are not easy and there is some work, e.g. by Kedlaya, to make them feasible.

Question 3 is false. You can easily write down smooth surfaces of any degree containing a straight line.

Off the top of my head, I don't have an example for your question 1, which is unconditional. But, if believe the ABC conjecture in its multiple summands variant, then diagonal hypersurfaces give examples (e.g. $x^n+2y^n+3z^n=6w^n, n$ large).

For question 2, you can look for surfaces of Picard number 1. One way to find such surfaces is to reduce modulo p and show that the reduction has Picard number 1 by computing its L-function. This will only work for odd degree. For even degree, you may need to compute the reduction modulo two primes with Picard number 2 but incompatible lattices so the Picard number of the original surface is 1 (this is a trick due to R. van Luijk). These computations are not easy and there is some work, e.g. by Kedlaya, to make them feasible.

Question 3 is false. You can easily write down smooth surfaces of any degree containing a straight line.

Off the top of my head, I don't have an example for your question 1, which is unconditional. But, if believe the ABC conjecture in its multiple summands variant, then diagonal hypersurfaces give examples (e.g. $x^n+2y^n+3z^n=6w^n, n$ large).

Edit: As Jason points out in the comments, my suggestion for question 2 is not enough. A different idea would be to use the idea for question 1. Namely use the function field ABC (which is a theorem) to show that a "fewnomial" (diagonal equations contain lines) equation has no curves of genus 0 or 1. I haven't worked out the details, though.

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created Aug 26, 2012 at 13:01

30.5k
6
85
151

For question 2, you can look for surfaces of Picard number 1. One way to find such surfaces is to reduce modulo p and show that the reduction has Picard number 1 by computing its L-function. This will only work for odd degree. For even degree, you may need to compute the reduction modulo two primes with Picard number 2 but incompatible lattices so the Picard number of the original surface is 1 (this is a trick due to R. van Luijk). These computations are not easy and there is some work, e.g. by Kedlaya, to make them feasible.

Question 3 is false. You can easily write down smooth surfaces of any degree containing a straight line.

Off the top of my head, I don't have an example for your question 1, which is unconditional. But, if believe the ABC conjecture in its multiple summands variant, then diagonal hypersurfaces give examples (e.g. $x^n+2y^n+3z^n=6w^n, n$ large).