Blowing up a subvariety - what can happen to the singular locus? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T02:54:08Z http://mathoverflow.net/feeds/question/73557 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/73557/blowing-up-a-subvariety-what-can-happen-to-the-singular-locus Blowing up a subvariety - what can happen to the singular locus? samian86 2011-08-24T11:57:49Z 2011-08-24T14:57:09Z <p>Let $X$ be a variety defined over a number field $k$. If I blow-up along some arbitrary subvariety of $X$, what are the possible outcomes for the dimension of the singular locus of the variety? If the subvariety lies outside the singular locus of $X$, then it stays the same, if it is carefully chosen, it might go down. Can it go up?</p> <p>To be more specific, my variety is a high dimensional hypersurface, and the subvariety I am blowing up is a linear space of much smaller dimension than the singular locus. I don't know if this changes the situation.</p> <p>I have a feeling this question might be more suited to stackexchange, but it didn't spark much interest over there <a href="http://math.stackexchange.com/questions/53676/blowing-up-a-subvariety-what-can-happen-to-the-singular-locus" rel="nofollow">http://math.stackexchange.com/questions/53676/blowing-up-a-subvariety-what-can-happen-to-the-singular-locus</a>. Apologies for wasting time if so.</p> http://mathoverflow.net/questions/73557/blowing-up-a-subvariety-what-can-happen-to-the-singular-locus/73569#73569 Answer by J.C. Ottem for Blowing up a subvariety - what can happen to the singular locus? J.C. Ottem 2011-08-24T14:57:09Z 2011-08-24T14:57:09Z <p>Any birational map $\pi:X'\to X$ is the blow-up of some ideal sheaf on $X$, so in general one must expect singularities on $X'$, even if the ideal is reduced (as you assume). </p> <p>As a concrete example, let $X=\mathbb{A}^n$ and blow-up the complete intersection subvariety gven by the ideal $I=(f,g)\subset k[x_1,\ldots,x_n]$. Then the blow-up of $X$ is the Proj of the Rees algebra $R[It]$ which is given by $k[x_1,\ldots,x_n,S,T]/(fS-gT)$. By choosing $f$ and $g$ appropriately one can produce varieties with singular locus of high dimension.</p> <p>For your specific example, when $Y$ is a linear space of small dimension, I don't know if the above can happen, but there are certainly cases where the dimension of the singular locus will be unchanged after the blow-up, (e.g when $Y$ a point on a singular surface).</p>