Let $f:\tilde{X} \to X$ be a normalization of projective variety. Let $L$ be a very ample line bundle on $X$. Is $f^\*L$ a very ample line bundle on $\tilde{X}$? If not true in general, is there any dimension restriction on $X$ for which this is true?



Does normalization of projective varieties preserve very ampleness
1) There are certainly many cases where $f^\*L$ is still very ample: e.g. if $X$ is an irreducible curve, any line bundle on $X$ of degree $\geq 2g(\tilde{X} )+1$ will have this property.


2) A typical situation where $f^\*L$ is not very ample is when $H^0(\tilde{X},f^\*L )\cong H^0(X,L)$. For instance, take for $X$ a curve with one ordinary double point $s$, and $L=\omega \_X(-p)$, $p$ a smooth point of $X$. Then $f^\*L=\omega \_{\tilde{X}}(s'+s''-\tilde{p} )$, where $s',s''$ are the 2 points of $\tilde{X} $ above $s$ and $\tilde{p} $ the point above $p$. Riemann-Roch gives $h^0(f^\*L)= g(\tilde{X} )=h^0(L)$ (I choose $p$ so that $h^0(s'+s''-\tilde{p} )=0$).


I think there is no simple criterion allowing to decide which situation occurs, you have to work.


