Dear J\'er\'emy, a nef divisor on a surface always has nonnegative selfintersection. For 9+n points the anticanonical has selfintersection -n, so it cannot be nef.

Small clarification to the last line: if you blow up more than 9 points, the anticanonical is not even nef.

@abx: ach, I should have thought a bit more before reposting. Anyway, it is good to know at least that you do not know the answer to the second variant. Thanks a lot!

By "miracle flatness", if f is not flat then X is not Cohen--Macaulay. But if x∈X is a non-CM point then I am very doubtful that the fibre through x can really be smooth (e.g. because locally near X the fibre is cut out by a regular sequence).

A useful reference might be "Galois groups of enumerative problems" by Joe Harris.

If you are unsure about which site is more appropriate for your question and want to try both, the accepted "best practice" is to post first on Math.SE, then wait for a reasonable amount of time, say a week; if after that time you don't get anything satisfactory, you can repost here. The point is that although crossposting might increase the probability of getting answers, it also increases the probability of people wasting their time by answering a question that has already been settled. Anyway, this is not my personal opinion, just an observation about the norms of these sites.

There is a consensus that one should not post the same question here and on Math.SE simulateneously.(For what it's worth, in this case I doubt you gain anything by having a copy of the question at MSE too.)

Oops, I should have said "birational" instead of "isomorphism". (But the point remains the same.)