All Questions
Tagged with blowup intersection-theory
9 questions
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How to correctly calculate the intersection number of the exceptional line with the canonical divisor of the blowup?
Let $X$ be a projective subvariety in $\Bbb P^{3}$ which is given by the equation $xyw-z^{3}=0$. Let $\widetilde{X} \rightarrow X$ be the blow up at the point $q :[x:y:z:w]=[0:0:0:1]$. There are two ...
- 31
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Surjectivity of the pushforward of a blowing-up morphism
Given a closed immersion $Z\rightarrow X$ and considering the blowing-up of $X$ at $Z$ and we get $\widetilde{X}$. I'm wondering is the proper pushforward on etale cohomology induced by the blowing-up ...
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Confusion about line bundles and the intersection product
Let $X = \mathbb{A}_{\mathbb{C}}^{2}$ and let $Y$ be the blowup of $X$ at the origin. Let $E \cong \mathbb{P}^{1}$ the exceptional divisor.
I think that we have a canonical inclusion $\mathcal{O}_{Y} \...
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Blow-up of a Pencil of Cubic Curves (Miranda's basic theory of elliptic surfaces)
In Rick Miranda's "The basic theory of elliptic surfaces" the Example (I.5.1) see page 7 on a pencil of plane curves contains an argument Inot understand yet:
Let $C_1$ be a smooth cubic ...
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Formula $m_p(C\cap D)=\sum_xm_x(C)m_x(D)$, for $x$ infinitely near $p$, applied to a concrete example
I'm reading this wikipedia article about infinitely near points.
In the section "Applications", the article says: If $C,D$ are irreducible curves on a smooth surface $S$ which intersect in a ...
- 10.5k
3
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Intersection Theory and Blow up
The following is from Fulton's Intersection Theory:
Theorem 6.7 (Blow-up Formula). Let $V$ be a $k$-dimensional subvariety of $Y$, and let $\widetilde{V} \subset \widetilde{Y}$ be the proper ...
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Self-intersection of a curve $C\subset \Bbb{P}^2$ after $3$ blow-ups
Let $P_0,P_1,P_2\in\Bbb{P}^2$ points in general position,consider the lines $\ell_i:=\overline{P_jP_k}$ for $\{i,j,k\}=\{0,1,2\}$ and the blow-up $\pi:S\to\mathbb{P}^2$ at $P_0,P_1,P_2$.
I was told ...
- 10.5k
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How to use Nakai-Moishezon criterion?
Theorem (Nakai- Moishezon): A Cartier divisor $D$ on proper scheme $X$ is ample if and only if, for every integral subscheme $Y$ of $X$, one has $D^{\text{dim}(Y)}. Y > 0$
Defintion: Let $X$ ...
- 269
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Self-Intersection under Blow up
Let $X$ be a algebraic surface, $\Lambda$ be a linear system, and $p\in \Lambda$ be its base point. We blow up $X$ at $p$, denote the exceptional curve $E$.
Q: For any $C\in|\Lambda|$, can we say $(...
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