Let $f:X\rightarrow Y$ be a proper surjective morphism. Suppose X have only normal crossing singularity. Let H and E are two irreducible components of the singular locus of X i.e, $Sing(X)=H\cup E$. So clearly H and E are codimension 1 sub varieties. Suppose $f|_{X\setminus E}$ is an isomorphism i.e f is isomorphism outside $E$. Also suppose $f|_E$ is a $\mathbb{P}^n$ bundle over $f(E)$. Is it possible that f is not a blow up along f(E)?
1 Answer
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Yes, it is possible.
Well, it is kind of cheap, but if $f$ is not projective, then there is no chance of it being a blow-up, so you might as well say that $f$ is projective. Also, I assume you mean $f(E)$ with the reduced subscheme structure.
Let $Y$ be the quintessential example of a non-Cohen-Macaulay surface: Take a smooth surface and glue two of its points "transversally". By this I mean that $Y$ along its only singular point would be analytically locally isomorphic to $Y'=Z(x,y)\cup Z(z,t)\subseteq \mathbb A^4_{x,y,z,t}$.
For the following computation this is what matters, so I will assume that $Y=Y'$. Blow up the singular point of $Y$: the two components (branches, really) separate and you get a $(-1)$-curve on each. Now glue the two surfaces together along their just created $(-1)$-curves to form a normal crossing singularity. This glued together "double" $(-1)$-curve is your $E$ and the glued surfaces together is $X$ (if you started with an irreducible $Y$, then $X$ will also be irreducible). If you must have an $H$ as well, then blow up another point on $E$, but $H$ is really a red herring.
Anyway, the original blow up morphism on each component gives a morphism $f:X\to Y$ which contracts $E\simeq \mathbb P^1$ to a single point, that is, to the singular point of $Y=Y'$. However, if you blow up that point, then the two components (branches) get separated, so $f$ is not the blow-up of $f(E)$.
If you really want a $\mathbb P^n$-bundle with $\dim f(E)>0$, then just take a product of all of this with your favorite smooth variety. If you want $n>1$, then do this with higher dimensional smooth varieties meeting in a single point.
References for your questions in the comments below:
- Gluing two surfaces along a curve should be pretty simple, but one could also just appeal to Karl Schwede's paper Gluing schemes and a scheme without closed points, especially Theorem 3.4 and Corollary 3.9.
- See [Hartshorne, III.7.17].
answered Jul 4, 2017 at 5:40
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$\begingroup$ Kovacs Thanks...actually this is almost like my setting. Anyway I have several questions: 1. when you say that glue the -1 curves, is it some general result that this gluing gives a scheme or surface in this case?( like for example it may be two copies of $\mathbb{P}^n$ which are being glued up.).It is not clear why it is a scheme in general. 2. Another thing is in your counter example is it possible that f is a blow up along a $\textbf{fractional ideal }$ supported on $f(E)$. Is there some general theorem of Grothendieck? Because I have heard people casually saying that...thanks $\endgroup$– user100841Commented Jul 4, 2017 at 14:15
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$\begingroup$ for 2. a reference will be very helpful if the answer is positive $\endgroup$– user100841Commented Jul 4, 2017 at 18:05
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$\begingroup$ Let $E_1$ and $E_2$ are two -1 curves in your example. Then $\mathcal{O}(-E_1-E_2)$ descends as a Cartier divisor to the glued scheme, which is trivial outside E. Actually this situation Gets complicated when there is another H intersecting E. In this case it is not clear whether $\mathcal{O}(-E_1-E_2)$ descends as a Cartier divisor. $\endgroup$– user100841Commented Jul 9, 2017 at 13:32
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