Ok, I’ve noticed that there were a few people following this blog. So I felt obligated to post something on a regular phase, like once a month or something… I doubt if I’ll be that diligent…

Recently I’ve been spending my spare time surfing on the Mathoverflow, on which I don’t understand most questions and answers(well, as always…). But I’ve seen one interesting question yesterday and gave it some thoughts, the website of the question with a beautiful answer is at here.

We know that every curve can be embedded into 3-projective space. So the question is given a field k, whether there is a surface S over k such that every smooth curve over k can be embedded in S. The original question is asked for k=C, the complex numbers, and Olivier Benoist gave an beautiful answer to this question using following strategy. Let me (shamelessly copy?) sketch HIS approach: suppose there is such a complex surface S, then we consider the Hilbert schemes of genus g curves in S, because Hilbert schemes have only countably many component, one of them (say, H_g) will dominate M_g (the coarse moduli of genus g curves) which is of general type (proved by Mumford, relies on the hypothesis that we are over C). So H_g is not rational, hence the map from H_g to Pic^0(S) (sending a divisor to its associated line bundle) is not trivial, which means the Albanese map of S is not trivial. Then we would easily come to a contradiction. Sorry, it’s a very brief sketch…

So now we may ask ourselves this question, what if k is not C? The argument above would go wrong in two ways. Firstly, the reason that H_g would dominate M_g is because C is uncountable, so countably many proper subvarieties of M_g couldn’t cover whole M_g. But what if k is the algebraic closure of Q? That is to ask do we have a surface containin every curves defined over number field? This probably would be a very difficult and admittedly very interesting question. Secondly, we also have to consider the situation of characteristic p. If k is algebraic closure of F_p(t), then we don’t know if M_g is unirational or not (Ok, probably M_g is not unirational.)? At least I think most of people would guess it’s not (Well, let me speak for myself…). This is also an interesting question to ask. Long time ago the only technique we had for this kind of question was introduced by Mumford. Now I think we have another way to try, the thing below is just brainstorm…

Recently Professor Voisin has a new method called decomposition of diagonal to attack this kind of question (not unirationess but some other birational properties of a variety), so would her method help? Also one may ask her/himself whether given a general genus g curve we can put it in a nontrivial family over P^1 or not?

Anyway, I just realized there are still lots of open questions (as always?) out there. This world is full of challenges and fun, huh.

If I may:

few people -> a few people

in a regular phase -> on a regular basis

I’ve spending -> I’ve been spending

an beautiful answer -> a beautiful answer

an complex surface -> a complex surface

many component -> many components

will dominant M_g -> will dominate M_g (I guess?)

a surface contain -> a surface containing

Long time ago the only technique we have -> Long time ago the only technique we had

Anyway, nice write-up!

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Haha, many thanks for helping my grammar! I’ll do this better next time ( I think ^_^)

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NP! If it starts to get uncomfortable please let me know ^^

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Nah, I need it.

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