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Does exist a smooth, complex, projective variety $X$ of dimension $d\geq2$ such that $X$ does not contain smooth, complex, projective curves of wichever genus $g\geq2$?

Answer by Bertie: No, it does not exists.

More in general, does previous statement hold for $X$ over any field of characteristic $0$?

Partial Answer: If the field is algebraically closed: no, it does not.

Open question: Does exist a smooth projective variety $X$ of dimension $d\geq2$ over a field $\mathbb{K}$ not algebraically closed of characteristic $0$, such that $X$ does not contain smooth projective curves of wichever genus $g\geq2$?

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    $\begingroup$ Could you please clarify: are you fixing $g$ and then asking for fixed $g$ whether there exists a smooth projective variety that contains no curve of genus $g$? The answer to that is yes, e.g., any simple Abelian variety of dimension $>g$. However, by the adjunction formula, the genera of complete intersection curves of sufficiently ample divisors are arbitrarily positive. So every projective variety of dimension $\geq 2$ contains curves of unbounded genera. $\endgroup$ – Jason Starr Feb 12 '17 at 13:42
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Every smooth projective variety of dimension at least 2 over an algebraically closed field contains curves of arbitrarily large genus.

(Choose an ample line bundle $L$ on $X$ and consider the embeddings $X \hookrightarrow \mathbf P^{N_d}$ given by tensor powers $L^d$; a general linear section of $X$ of the appropriate codimension is then a smooth curve of genus $g(d)$, where $g(d)$ grows without bound as a function of $d$.)

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    $\begingroup$ As usual, Jason Starr got there before me... $\endgroup$ – Bertie Feb 12 '17 at 13:48
  • $\begingroup$ Thank you!, and over a field of characteristic $0$ and not algebraically closed? $\endgroup$ – Armando j18eos Feb 12 '17 at 14:33
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    $\begingroup$ @Armandoj18eos This argument works over any infinite field, and with Poonen's Bertini theorem will work over finite fields. $\endgroup$ – Will Sawin Feb 12 '17 at 19:25

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