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Colin,

Francesco has already mentioned the "embedding theorem for big line bundles" that you were probably looking for, but I would like to add that this is pretty much the definition of big.

An alternative point one could make is that according to Kodaira's Lemma "big=ample+effective", so a high power of a big divisor is very ample on the complement of an effective divisor.

As for your last question, if $f:Y\to X$ is any proper morphism and $\mathscr L$ is an ample line bundle on $X$, then the image of if $g$ denotes the morphism(!) (say $g$) defined by the global sections of very high powers of $f^*\mathscr L$is , then it follows that $X$. f$factors through$g$. In particular, then$g$is an isomorphism to its image (hence an embedding) wherever$f$is (if$f$is not birational, then this "wherever" is the empty set). 3 edited body; added 7 characters in body; added 55 characters in body Colin, Francesco has already mentioned the "embedding theorem for big line bundles" that you were probably looking for, but I would like to add that this is pretty much the definition of big. An alternative point one could make is that according to Kodaira's Lemma "big=ample+effective", so a high power of a big divisor is very ample on the complement of an effective divisor. As for your last question, if$f:X\to Y$f:Y\to X$ is any proper morphism and $\mathscr L$ is an ample line bundle on $Y$, X$, then the image of the morphism(!) (say$g$) defined by the global sections of very high powers of$f^*\mathscr L$is$Y$. X$. In particular, if $f$ is birational, then it $g$ is an isomorphism to its image (hence an embedding) wherever $f$ is (if $f$ is not birational, then this "wherever" is the empty set).

2 added 27 characters in body

Colin,

Francesco has already mentioned the "embedding theorem for big line bundles" that you were probably looking for, but I would like to add that this is pretty much the definition of big.

An alternative point one could make is that according to Kodaira's Lemma "big=ample+effective", so a high power of a big divisor is very ample on the complement of an effective divisor.

As for your last question, if $f:X\to Y$ is any morphism and $\mathscr L$ is an ample line bundle on $Y$, then the image of the morphism(!) defined by the global sections of very high powers of $f^*\mathscr L$ is $Y$. In particular, if $f$ is birational, then it is an isomorphism to its image (hence an embedding) wherever $f$ is.

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