The blow-ups tag has no wiki summary.
2
votes
1answer
139 views
Blowing-up the Grassmannian at a point
Does anyone know what the blow-up of the Grassmannian at a point looks like? Consider $G=Gr(r,n)$ and $V\in G$. I want to understand more explicitly what $Bl_V(G)$ should mean.
Of course for affine ...
3
votes
0answers
111 views
Blowig-up a point in the singular locus
Let $X\subset\mathbb{P}^n$ be a variety singular along a smooth subvariety $Z\subset X$ of positive dimension. Let us assume that $X$ has ordinary singularities along $Z$. Now, let $\pi:Y\rightarrow ...
4
votes
0answers
159 views
Quadrics and Moduli Spaces
It is well known that $\overline{M}_{0,5}$, the moduli space of $5$-pointed rational curves, can be realized as the blow-up of $\mathbb{P}^2$ in four general points. Therefore, we may interpret ...
2
votes
1answer
314 views
A question about an intersection number
Let $\pi:Y\rightarrow \mathbb{P}^3$ be the blow-up of two points $p,q\in\mathbb{P}^3$, and then of the strict transform of the line $L$ spanned by them. Now, Let $E_p,E_q, E_{p,q}$ be respectively the ...
4
votes
0answers
82 views
Blow-up of $\mathbb{P}^4$ along a smooth surface
Let $\pi \colon X\to \mathbb{P}^4$ be the blow-up of a smooth surface $S\subset \mathbb{P}^4$. Is there a formula to compute $(K_X)^4$ ? (which should be dependent on invariants of $S$).
In dimension ...
0
votes
1answer
111 views
Intersection Matrix of a resolution
Probably this is a very easy question. Let $f:X\rightarrow S$ be a resolution of a projective surface such that
$$K_X = f^{*}K_S+\sum_ia_iE_i$$
with $a_i>0$. By Grauert-Mumford theorem the ...
6
votes
4answers
645 views
Cone over the Veronese surface
Let $V\subset\mathbb{P}^5$ be the Veronese surface and let $X\subset\mathbb{P}^6$ be the cone over it. Since $X$ is $\mathbb{Q}$-factorial there are two integers $a,b$ such that $aK_X = ...
8
votes
2answers
348 views
Blow-ups of $\mathbb{P}^{n-3}$ and $(\mathbb{P}^1)^{n-3}$
Let us consider the points
$$p_1=[1:0:...:0],p_2 = [0:1:...:0],...,p_{n-2} =[0:...:0:1],\\
p_{n-1}=[1:1:...:1]\in\mathbb{P}^{n-3}$$
and the blow-up $X = Bl_{p_1,...,p_{n-1}}\mathbb{P}^{n-3}$.
...
3
votes
2answers
177 views
Standard plane Cremona transformation
Let us consider nine general points $p_1,...,p_9\in\mathbb{P}^2$ and the line $L = \left\langle p_1,p_2\right\rangle$. Take the standard Cremona $f_1$ centred in $p_3,p_4,p_5$, then $C_1 = f_1(L)$ is ...
1
vote
0answers
101 views
A question on the secondary fan
I am studying the secondary fan decomposition of the effective cone of a projective variety $X$. Let as assume that $X$ is a Mori Dream Space. As far as I understand passing from a cone of maximal ...
1
vote
0answers
109 views
Projective tangent cones, ordinary singularities and blow-ups
Let $X\subset\mathbb{P}^n$ be a projective variety and let $Y\subset X$ be the singular locus of $X$. Assume that $Y$ is smooth. I would like to know if the following are equivalent:
$X$ has an ...
1
vote
0answers
113 views
Resolution of singularities of projective varieties
Let $X\subset\mathbb{P}^n$ be an irreducible variety, and let $Sing(X)$ be its singular locus. Let $Y$ be the blow-up of $\mathbb{P}^n$ along $Sing(X)$. Assume that we know that the strict transform ...
2
votes
0answers
175 views
When is the proper transform of a subscheme equal to the total transform?
Say that $X \subset Y \subset \mathbb{P}^n$ are closed subschemes and let $\pi: \tilde{Y} \rightarrow Y$ be the blowup along $X$.
If $H$ is a hyperplane section of $Y$, under what conditions (on ...
1
vote
2answers
234 views
When the contraction is a morphism defined over $\overline{\mathbb Q}$
Suppose that $S$ is a complex projective surface defined over $\overline{\mathbb Q}$, namely there exists a surface $S_{\overline{\mathbb Q}}$ over $\overline{\mathbb Q}$ such that:
...
1
vote
1answer
226 views
Blow-ups and cohomology
I'm trying to understand how to compute the Chow ring of a blow-up.
Let $W\subset \mathbb P^4$ be a smooth surface and let $X$ be the blow-up of $\mathbb P^4$ along $W$ with exceptional divisor $E$. ...
3
votes
1answer
126 views
A question on klt pairs
Let $D$ be a $\mathbb{Q}$-divisor in a smooth variety $X$. In Lazarsfeld book "Positivity in Algebraic Geometry 2" I found Proposition 9.5.13 saying that if for any $x\in D$ we have $mult_xD < 1$ ...
-1
votes
2answers
116 views
Blow up solution for a Riccati's equation
I apologize for the problem too simple, but I'm not able at solving it.
Consider the Cauchy problem
$$
\left\{
\begin{array}{l}
\dot x=x(t)^2+t\\
x(0)=0
\end{array}
\right.
$$
Show that its solution ...
2
votes
2answers
234 views
Holomorphic vector fields on blow-ups of CP^2
On $X=CP^2\#k{(-CP^2)}$ in $k$ generic points, let $h^i=\dim H^i(T^{1,0}X)$, for $i\ge 0$. First, we know $h^i=0$ for $i\ge 2$. By Riemann–Roch formula, I obtain that $h^0-h^1 = 8-2k$. Would someone ...
2
votes
0answers
125 views
A question on resolution of singularities
I am wondering if it could be possible in particular cases to resolve a singularity of dimension $n$ by blowing-up a locus of dimension smaller than $n$.
For instance consider a cubic surface ...
4
votes
2answers
273 views
Cohomology of the toric variety $X_\Sigma=\mathbb C^2\sqcup \mathbb C^2\big/\left((x,y)_1\sim(x^{-1},y^{-1})_2\right)_{x,y\neq 0}$
I'm writing a thesis on Chow rings of toric varieties and am looking for a reference on the singular cohomology ring of the blowup of $\mathbb C^2$ at xy=0, i.e. at the coordinate axes. The ...
0
votes
1answer
239 views
Projection of a hypersurface from a point
Let $k$ be an algebraically closed field. We consider the projective space $\mathbb P_n$ over defined over $k$, the point $Q=(0:\dots:1)$, the hyperplane $H=\{X_n=0\}$ and a hypersurface $X$. We want ...
2
votes
1answer
160 views
Singularities of secant varieties of rational normal curves
Let $C\subset\mathbb{P}^n$ be a rational normal curve of degree $n$, and let $Sec_k(C)\subset\mathbb{P}^n$ be its $k$-th secant variety. By Theorem 1.1 in this paper:
...
1
vote
2answers
202 views
Small birational maps and singularities of the pair
Let $f:X\dashrightarrow Y$ be a small birational map, where $X,Y$ are normal $\mathbb{Q}$-factorial varieties. Let $\Delta_X\subset X$ be an effective $\mathbb{Q}$-divisor such that the pair ...
2
votes
0answers
86 views
Jacobian Conjecture, Cubic-Keller maps
I have recently read an interesting article about the Jacobian Conjecture, in particular the reduction to the case $f(x) = x + A(x)^3$.
I was wondering about codimension one divisors on $Y = A^n$. ...
1
vote
1answer
144 views
Ample divisors on $\mathbb{P}^n$ blown-up at $k$ general points
Let $X$ be the blow-up of $\mathbb{P}^n$ at $k$ general points. We can assume $k\leq n+4$. Let
$$D = aH-b_1E_1-...-b_kE_k$$
be a divisor on $X$. Are there conditions on $a,b_1,...,b_k$ ensuring that ...
0
votes
1answer
375 views
Simple normal crossing divisors
I found the following definition.
A Weil divisor $D = \sum_{i}D_i \subset X$ on a smooth variety $X$ is simple
normal crossing if for every point $p \in X$ a local equation of $D$
is ...
-1
votes
1answer
149 views
Singular irreducible quadrics
Let $Q\subset\mathbb{P}^n$ be the quadric hypersurface defined by
$$x_0^2+x_1^2+...+x_k^2 =0.$$
If $2\leq k\leq n-1$ then $Q$ is irreducible and $Sing(Q)$ is a linear space of dimension $n-k-1$.
If ...
0
votes
3answers
311 views
Higher cohomology of sheaves on a projective space
Let $S\subset\mathbb{P}^n$ be a finite set of $s$ reduced points. Let $\mathcal{I}$ be the ideal sheaf of $S$ in $\mathbb{P}^n$. We consider the sheaf
...
3
votes
1answer
162 views
On Neron-Severi group of normal projective surfaces and blow up
Let $X$ be a normal projective surface with at most rational singularites (in finitely many points). Let $\pi:\tilde{X} \to X$ be the blow up of $X$ at finitely many singular points. The question is ...
2
votes
2answers
313 views
blow-up of $\mathbb{P}^5$ as a projective bundle
I am wondering if the blow-up of $\mathbb{P}^5$ along three disjoints $\mathbb{P}^1$ (say in generic position) can be understood as a projective bundle over some nice (Fano?) variety.
If one ...
3
votes
1answer
216 views
Blow-up of $\mathbb{P}^4$ along a quadric surface
Let $Q\subset\mathbb{P}^3\subset\mathbb{P}^4$ be a smooth quadric surface, and let $X = Bl_Q\mathbb{P}^4$ the blow-up of $\mathbb{P}^4$ along $Q$. Let $H$ be the pull-back of the hyperplane section of ...
5
votes
1answer
550 views
Top self-intersection of exceptional divisors
Let $Y\subset\mathbb{P}^n$ be a smooth variety of codimension two. Consider the blow-up $X = Bl_Y\mathbb{P}^n$ of $\mathbb{P}^n$ along $Y$, and let $E$ be the exceptional divisor over $Y$. Then $E$ ...
1
vote
0answers
84 views
Can discrepancy change after pseudo isomorphisms?
Let $X$ and $Y$ be two projective irreductive algebraic varieties of dimension $3$ and let $f:X\dashrightarrow Y$ be a pseudo isomorphism, i.e. a birational map which restricts to an isomorphism ...
2
votes
2answers
165 views
blow-up along non-pure dimensional subvarieties
When speaking about blow-ups, I've seen that everybody says "let $Y \subset X$ be a smooth closed subvariety of codimension $r$..."
What happens when one want to blow-up subvarieties which are not of ...
0
votes
1answer
75 views
is the exceptional divisor of an equivariant blow-up linearized?
I hope someone can help me with this.
Let $X$ be a smooth projective variety, say over the complex numbers and let $Z$ be a closed subvariety of $X$. Assume that $X$ is acted upon by a finite group ...
4
votes
1answer
175 views
Blowing up rational singularities
Let $X$ be a projective surface embedded into $\mathbb{P}^n_{\mathbb{C}}$ having at most rational singularities. Let $\tilde{X} \to X$ be the minimal resolution of $X$. Is it possible to embed ...
5
votes
1answer
298 views
which varieties can appear as exceptional divisors?
Let $\pi$ be the blow up of $\mathbb{A}^n$ at an ideal
$I_{p}$ which is supported at a close point $p$; let $E$ be the associated exceptional divisor. What are the possible varieties $Y$ that can ...
3
votes
1answer
188 views
Is there a blow-up formula for the derived category of a singular ambient variety?
For a nonsingular variety sitting inside a nonsingular ambient variety there is a semi-orthogonal decomposition of the derived category of the blow-up (with center that subvariety).
What can be said ...
1
vote
1answer
334 views
Cohomology of tangent bundles
Let $X$ be a smooth scheme and $Z\subset X$ a smooth subscheme. Consider the blow-up
$$\pi:\widetilde{X}:=Bl_{Z}X\rightarrow X$$
of $X$ along $Z$.
What is the relation between the cohomology of the ...
4
votes
1answer
389 views
Is blowing down functorial?
Let $f: X \to Y$ be a morphism of smooth surfaces and $C \subset Y$ be a contractible exceptional curve on $Y$. Assume that $f^{-1}(C)$ is a contractible exceptional curve as well. Denote by $X'$ ...
3
votes
2answers
352 views
Terminology for blowups in algebraic geometry
This is a partial duplicate of this Stack Exchange question which unfortunately got no answer.
All schemes are Noetherian and of finite type, although they need not be normal.
With $Z \subset X$ a ...
1
vote
1answer
271 views
blow-up along singular variety
Can somebody give me a nice example of blow-up of a smooth algebraic variety along a singular subvariety? Something I can do some exercise on and check the differences with a smooth blow-up. Thanks!
1
vote
0answers
153 views
Contractibility of union of smooth rational curves in famillies
Let $\pi:\mathcal{X} \to S$ be a family of smooth surfaces containing a subfamily $\mathcal{Y} \subset \mathcal{X}$ proper, flat over $S$ parametrizing contractible curves in $\mathcal{X}$ satisfying ...
6
votes
0answers
185 views
blowups and group actions
Let $X$ be a smooth projective variety over the complex numbers and assume that $X$ is equipped with the action of a finite group $G$.
Denote by $Z$ the closed subscheme of fixed points of $G$ and ...
1
vote
0answers
408 views
cohomology of a blowup: reference needed
Does anybody know a reference in which the computation of the cohomology of a blow-up is made in detail?
0
votes
2answers
262 views
blow-ups and singularities
Let $X$ be a smooth and projective variety over a field of characteristic zero. Let $Y$ be a normal variety, with finite quotient singularities (an orbifold!) and let $\pi: Y \to X$ be a finite ...
4
votes
1answer
152 views
$f^{-1}\mathcal I \cdot \mathcal O_X$ vs $f^\ast \mathcal I$
Let $X$ ad $Y$ be (noetherian) schemes and let $\mathcal I \subseteq \mathcal O_Y$ be a sheaf of ideals on $Y$. Let $f \colon X \to Y$ be a morphism of schemes. In general the sheaf $f^\ast \mathcal ...
3
votes
1answer
291 views
blow-up and normal crossings
Hi friends,
Can anybody help me with the following question?
I start with a projective smooth variety $X$ over a field $k$ of a characteristic 0 and $D$ a normal crossing divisor on $X$, with ...
0
votes
0answers
108 views
Is the modification a rational map?
Good morning,
I would like to ask the following question concerning the desingularisation, but I'm not familiar at all with these notions.
We have the following theorem of Hironaka: Let $X\subset ...
3
votes
0answers
298 views
Blowing up a projective surface
Let $X$ be a smooth degree $d$ ($d>5$) surface in $\mathbb{P}^3$. We now blow up $X$ at a point, embed it in some projective space, and and consider a projection of it into $\mathbb{P}^3$. The ...
