When do blow-up and quotient commute? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T16:24:40Z http://mathoverflow.net/feeds/question/105996 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105996/when-do-blow-up-and-quotient-commute When do blow-up and quotient commute? Jerome 2012-08-30T21:48:24Z 2012-08-31T04:40:31Z <p>Let a finite group $G\subset SL(n,\mathbb{C})$ act on $\mathbb{C}^{n}$ in a natural way. Assume there is a crepant resolution of $f:X\rightarrow \mathbb{C}^{n}/G$. When is it possible to write $X$ as $Y/G$ for some $Y$ and $G$-action on $Y$? </p> <p>This is true for example $\pm id_{\mathbb{C}^{2}} \subset SL(2,\mathbb{C})$ acting on $\mathbb{C}^{2}$. Is it still true for example $$\langle diag(e^{\frac{2\pi i}{3}},e^{\frac{2\pi i}{3}},e^{\frac{2\pi i}{3}})\rangle \cong \mathbb{Z}/3\mathbb{Z} \subset SL(n,\mathbb{C})$$ acting on $\mathbb{C}^{3}$? FYI, this has toric crepant resolution. </p> http://mathoverflow.net/questions/105996/when-do-blow-up-and-quotient-commute/106015#106015 Answer by Sasha for When do blow-up and quotient commute? Sasha 2012-08-31T04:40:31Z 2012-08-31T04:40:31Z <p>By definition $C^n/G = Spec(C[x_1,\dots,x_n]^G)$ and $X$ being a blow up of an ideal $I$ on $C^n/G$ can be written as $$X = Proj_{Spec(C[x_1,\dots,x_n]^G)}(C[x_1,\dots,x_n]^G \oplus I \oplus I^2 \oplus \dots).$$ Now assume the ideal $I \subset C[x_1,\dots,x_n]^G$ can be written as $$I = J \cap C[x_1,\dots,x_n]^G = J^G,$$ where $J \subset C[x_1,\dots,x_n]$ is a $G$-invariant ideal (the simplest thing to do is to take $J = C[x_1,\dots,x_n]\cdot I$). Take $$Y = Proj_{C[x_1,\dots,x_n]}(C[x_1,\dots,x_n] \oplus J \oplus J^2 \oplus\dots).$$ Then $Y/G = X$. </p>