All Questions
Tagged with blowup commutative-algebra
21 questions
2
votes
0
answers
337
views
Understand the Proj construction and blow-up
I have some problem understanding the Proj construction. I hope I can understand it better by the following example:
Let $I\subset R$ be an ideal, and consider ${\rm Proj}(\oplus I^k)$, which is the ...
- 1,111
1
vote
1
answer
74
views
About the integral closure of a DVR inside a galois extension of function field with two variables
Edit:
I see my mistake now (murphy),
w(X + Y) = min(w(X), w(Y)) only when the valuations are not the same.
I have the following algebraic problem, which I encountred after thinking about blowups.
My ...
- 39
2
votes
0
answers
171
views
Hartshorne Theorem II.7.17 - Why is $\mathscr{I} \to \mathscr{M}^{-dn}f_*\mathscr{L}^d$ injective?
I'm reading the proof of the following theorem from Hartshorne (Theorem II.7.17), which says the following:
Let $Z$ be a variety and let $X$ be a quasi projective variety, both over $k$. Suppose $f:Z\...
- 851
2
votes
0
answers
121
views
The Jacobian ideal under Blow up
Let $X$ be a singular hypersurface in $\mathbb{C}^n$ defined as the zero locus of $f(x_1, \dots, x_n)$ and denote $Jac(f)=\langle \partial_1(f), \dots, \partial_n(f) \rangle$, where $\partial_i(f)$ is ...
3
votes
1
answer
339
views
Ideal sheaf of a birational morphism arising from successive blowups
Let $L\subset \mathbb C^3$ be the line defined by $x=y=0$, and $p\in L$ the point defined by $x=y=z=0$. Let's consider the blowup of $\mathbb C^3$ at $p$ and then blow up the strict transform of $L$ ...
- 4,804
1
vote
1
answer
307
views
Projection from Blowup is Isomorphism Away from Exceptional Set
I'm following Eisenbud's description of blowing up: let $X$ be an affine algebraic variety, $R$ the coordinate ring of $X$, and let $a_1,\ldots,a_r$ generate $R$ as a $k$-algebra. Let $Y\subseteq X$ ...
- 1,383
1
vote
1
answer
46
views
On the intersection of affine open with the blowing-up of $\Bbb C^n$ at the origin
the blowup of $\Bbb C^n$ at the origin is the subvariety of $ \Bbb P^{n-1} \times \Bbb C^n$ given by
$B = V(x_{i-1} y_j - x_{j-1} y_i \mid 1 \le i < j \le n)$.$\ \ \ $ (1)
I am interested ...
- 3,312
1
vote
0
answers
77
views
Symbolic Rees Algebra of an ideal in a Noetherian excellent ring
For an ideal $I$ in a commutative Noetherian ring $R$ and integer $n\ge 0$, the $n$-th symbolic power of $I$ is define as $I^{(n)}:=\cap_{P\in Ass(R/I)} \phi_P^{-1} (I^nR_P)$ , where $\phi_P : R\to ...
- 1,682
1
vote
1
answer
635
views
Exceptional divisor of the blow-up of affine cone at the vertex
Let $f(x) \in \mathbb{C}[X_1,...,X_n]$ be a homogeneous polynomial in $n$ variables such that the zero locus $V$ of $f$ in $\mathbb{C}^n$ is singular only at the origin. Denote by $\pi:\widetilde{V} \...
- 397
0
votes
0
answers
251
views
Blow up algebras
Consider a ring $R$, an ideal $I$ in $R$ (with generators $g_1,\ldots,g_n$). Then for some $f\in I$ the map $R\rightarrow R[x_1,\ldots,x_n]/(fx_1-g_1,\ldots,fx_n-g_n)$ should correspond to blowing up $...
- 470
1
vote
0
answers
66
views
blow up of affine reducible variety at point has isomorphic total ring of fractions?
Assume $X$ is an reducible affine variety over $\mathbb{C}$ with coordinate ring $A$. Let $\pi \colon \tilde{X} \to X$ be the blowing up at a point $p \in X$ (which can be included in more than one ...
- 394
3
votes
0
answers
432
views
Blow up at point is finite?
Let $X$ be an affine algebraic curve with $0 \in X$ and $\tilde{X}$ the strict transform of $X$ w.r.t the blowup of $X$ at $0$. How to prove that $\pi \colon \tilde{X} \to X$ is finite? Is it even ...
- 394
3
votes
0
answers
326
views
Blow up of plane curve is Normalization of local ring?
I have a question concerning normalizations of plane curves, which I know little about.
Consider the simple node $V(f = y^2 - x^3 - x^2)$. Then $t=y/x$ is integral over $k[x,y]$ so that $(k[x,y]/f) \...
- 394
4
votes
1
answer
161
views
Calculation of valuation ring of a valuation associated to a blowup
Let $\mathfrak{m} = (x,y) \subset k[x,y]$. Then the valuation $v$ of $k(x,y)$ associated to the exceptional divisor of the blowup should be defined by $$v(f) = \mathrm{sup}(n|f \in \mathfrak{m}^n), f\...
- 1,054
5
votes
0
answers
167
views
Affine cover of blow-up along ideal
I would like to find affine cover of blow-up $X = Bl_{I} \mathbb A^2$, where $I=(x^3, xy, y^2)$. I know that $X=\{((x_1, x_2),[y_1,y_2,y_3])\subset \mathbb A^2\times\mathbb P^2: x_1^3y_2=x_1x_2y_1, ...
- 275
0
votes
2
answers
740
views
Is the dual of the normal bundle of the exceptional divisor ample
Let $X$ be a non-singular projective variety and $Y \subset X$ a non-singular subvariety. Denote by $\tilde{X}$ the blow-up of $X$ along $Y$ and $E$ the associated normal bundle. Denote by $N_{E|\...
- 399
0
votes
0
answers
221
views
Blow up along a valuation
What does it mean to blow up along a valuation? Under what circumstances do we usually blow up along a valuation?
I am having trouble finding a precise definition, so any references or examples ...
- 421
3
votes
1
answer
391
views
Blowing up an affine scheme at a regular point
I am reading Liu's Algebraic Geometry and Arithmetic Curves and get stuck at Lemma 8.1.2:
Let $A$ be a Noetherian ring an define for an ideal $I \subset A$ the $A$-algebra
$$\tilde{A}:=\bigoplus_{d\...
- 165
1
vote
0
answers
328
views
Description of Blow up via Ress Algebra
Let $X=\mathrm{Spec}(R)$ be an affine scheme and $Z=\mathrm{Spec}(R/I)$ be a closed subscheme of $X$ defined by an ideal $I$ of $R$. The blow up of $X$ along $Z$ is (say)$\mathrm{Bl}_{Z}(X)=\mathrm{...
- 5,209
3
votes
1
answer
206
views
Associated graded ring of a Fermat cubic
Let $R$ be a graded Fermat cubic, i.e. $R$ is a graded ring given by
$$
R=\mathbb{C}[x,y,z]/(x^3+y^3+z^3),
$$
with a standard grading $\operatorname{deg}(x)= \operatorname{deg}(y)=\operatorname{deg}(...
- 6,589
166
votes
1
answer
5k
views
What is the Picard group of $z^3=y(y^2-x^2)(x-1)$?
I'm actually doing much more with this affine surface than just looking for the Picard group. I have already proved many things about this surface, and have many more things to look at it, but the ...
- 1,693