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Why study projective normality of a variety ? What are the applications ? How does it relate to non-singularilty, rationality etc of the variety ?

user26857
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Jack
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    One nice thing, and I think I learned this from reading someone of Lazarsfeld's, is this: say you have a curve $C$ and then an embedding given by a line bundle $L$. It's very natural to ask about the number of hypersurfaces in $\mathbb{P}(H^0(L))$ of a given degree $d$ on which $C$ lies. If your embedding is projectively normal then you can read this off of $H^0(L^{\otimes d})$ and probably use Riemann-Roch. – Hoot Jun 24 '15 at 16:09
  • The same question (almost verbatim) was asked and answered here: http://math.stackexchange.com/questions/409222/projective-normality although Georges did not say anything about applications. Note that projective normality is a property of an embedding, not a variety, so it is unlikely to have much relation with the properties you mention, which are intrinsic. About the only thing one can say is that projectively normal by definition implies normal, which in some situations is a weaker substitute for nonsingularity. – Relapsarian Jun 24 '15 at 22:20
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    A surprising and powerful application is, if $X\subset\mathbb{P}^n$ with $n\geq 6$ and $X$ smooth of codimension 2, then it is a complete intersection if and only if it is projectively normal. This is closely related to what is known as Hartshorne's conjecture. – Mohan Jun 25 '15 at 15:59
  • @Mohan: When you say "$X$ smooth of codimension 2", do you mean the singular locus of $X$ has codimension $\geq2$ in $X,$ or do you mean $X$ is smooth and has codimension $2$ in $\mathbb{P}^{n}$ (so $\dim{X}=n-2$)? – Will R Mar 22 '21 at 02:35
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    @Will R Codimension two in the projective space . $X$ is smooth . – Mohan Mar 22 '21 at 14:55

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