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Example of projective variety that do not contain algebraic curves of genus strictly greater...
Every smooth projective variety of dimension at least 2 over an algebraically closed field contains curves of arbitrarily large genus. … 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 …