# Questions tagged [blow-ups]

The tag has no usage guidance.

168 questions
Filter by
Sorted by
Tagged with
72 views

### Intersections of strict transform and strict transform of intersections

Let $Z_1,Z_2$ and $Y$ be subvarieties of a locally complete intersection variety $X$ over $\mathbb C$. Consider the strict transforms of $Z_1$ and $Z_2$ in the blowup $Bl_YX$, the question is: when ...
60 views

### Is the blowup of a toric variety corresponding to a subdivision normal?

Toroidal Embeddings 1 by KKMS say subdividing the fan of a toric variety yields the fan of a normalized blowup. How do I avoid normalization? Do I need to choose my subdivision carefully, or is it ...
171 views

### Surgery for algebraic varieties

I have a number of vague questions that I wasn't sure whether they are suitable to ask or not, but I decided to ask! According to this result, any two birational varieties can be constructed by a ...
106 views

### Factorization of birational maps in char $p$

So I was reading about the factorization result that any birational map between smooth varieties is composition of blow-ups and blow-downs with smooth centers. It is apparently true only in ...
154 views

207 views

### Strict transform of a tangent curve under blow-up

$\DeclareMathOperator{\Bl}{\operatorname{Bl}}$It is known that if we have a projective variety $X$ and a projective smooth subvariety $Y$ then the exceptional divisor $E \subset \Bl_{Y}X$ of the blow-...
98 views

100 views

### Determining if a morphism is a blowup along a given subvariety

Let $X,\tilde{X}$ be two smooth projective varieties over $\mathbb{C}$, and let $\pi:\tilde{X}\rightarrow X$ be a projective morphism. Let us moreover assume that there exists a smooth closed ...
70 views

### Properties of the contraction map of the exceptional divisor of a blow-up

Let $X$ be a smooth variety and $Z$ be a smooth subvariety. Consider the blow-up at $Z$ $$\pi:\mathrm{Bl}_Z(X)\rightarrow X$$ and let $E$ be the exceptional divisor. What are the properties of the ...
197 views

### Picard group modulo codimension 2

Let $X$ be a normal (possibly singular) projective surface over $\mathbb{C}$. Consider the set $M_X$ of all coherent sheaves $F$ on $X$ such that there exists a finite subset $Y\subset X$ such that $F$...
291 views

### Simple proof that the arithmetic genus is non-negative

I take an irreducible and reduced closed curve $C\subseteq \mathbb{P}^n$, defined over an algebraically closed field $k$ and define the arithmetic genus $p_a(C)$ as the integer such that the Hilbert ...
69 views

187 views

### Very ample divisors on blow ups of the projective plane

Let $X$ be $\mathbb{P}^2$ blown up at $k$ points in general position. The Picard group of $X$ is just $\mathbb{Z}^{k+1}$ and one knows the intersection product explicitly. If $D$ is an ample divisor, ...
60 views

### Blowing up the base of an elliptically fibered (non Weierstrass) threefold

Suppose $X$ is an elliptically fibered smooth threefold, with a nontrivial Mordel-Weil group. Lets call the sections $\sigma_i$ ,$i=1 \dots n$. None of these "sections" are honestly a section, they ...
154 views

### Gradient blowup for the 1-dim heat equation near an irregular boundary point

I'm trying to estimate the rate of boundary gradient blow-up for the 1-dim heat equation near an irregular boundary point. Let $b(t) := (1-t)^\alpha$, $\alpha < 1/2$. Let $u(t,x)$ solve the ...
240 views

### The Chow ring of a blow-up along a badly embedded subscheme

Let $X$ be a variety and $i:Y\hookrightarrow X$ be a regularly embedded closed subvariety, and write $\tilde X$ for $Bl_Y X$ the blow-up of $X$ along $Y$. Let $A^*(X)$ refer to the Fulton-Macpherson ...
650 views

I would like to work through computing the blow-up of a particular curve along a subvariety consisting of just two points, both of which are ordinary double points. Let $$F(X,Y,Z) = X^4 + Y^4 - X Y^2 ... 2answers 887 views ### Topology of the blowup of a surface at a point (connected sum) Let S be a complex algebraic (smooth) surface and \widetilde{S} be the blowup of S at a point p\in S. I would like to understand the statement: As a topological manifold, \widetilde{S} ... 0answers 48 views ### How can I describe in explicit geometric terms the (in general non-complete) linear system? Let \varphi\!: S \to S^\prime be a birational morphism of projective non-singular irreducible surfaces over a (algebraically closed) field k and let D \in \mathrm{Div}(S). Also, let \big(\... 0answers 197 views ### When is the strict transform of very ample divisor ample? Let X be a projective variety and Y \subset X is a very ample divisor on X. Let Z \subset X be a regular subvariety (but X need not be regular along Z) of codimension 2 and \pi:\tilde{X}... 1answer 124 views ### Gysin map for projective sub-bundles of exceptional divisors Let X be a smooth, projective variety, Y \subset X a smooth, projective subvariety of codimension 3. Denote by \pi:\tilde{X} \to X the blow-up of X along Y and by E the exceptional ... 0answers 610 views ### Cotangent sheaf of a blow up It's well known that for a nonsingular variety X and a nonsingular subvariety Y with \text{codim}(Y,X) \ge 2, we can compute the canonical sheaf of \widetilde{X} (the blow up of X along Y) ... 1answer 161 views ### Blowing-up an ideal generated by squares Let f_1,\dots,f_r be regular functions on a smooth projective variety X, and consider the ideals I = (f_1^2,\dots,f_r^2) and J = (f_1,\dots,f_r). Let Y = Z(I) and W = Z(J) be the ... 0answers 441 views ### Cohomology of a blow-up of a real algebraic variety Let X be a complex algebraic variety, Z \subset X a closed subvariety, \mathrm{Bl}_Z X the blow-up and E the exceptional divisor. There is an isomorphism of cohomology groups$$ H^k(X(\mathbf ...
Let $X$ be a non-singular, projective variety and $Y \subset X$ non-singular, projective subvariety. Denote by $\pi:W \to X$ the blow-up of $X$ along $Y$. Let $Y_1 \subset W$ be a non-singular ...
Let $C$ be a curve in a smooth $3$-fold $X$ with an ordinary node $p\in X$. Blow-up $p$ let $E$ be the exceptional divisor, and $\widetilde{C}$ the strict transform of $C$. Furthermore let $L$ be the ...