Questions tagged [intersection-theory]
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376 questions
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Geometric examples of the Serre intersection formula
The Serre intersection formula, as an alternating sum of contributions from Tor-groups, is something that combines a lot of ingredients that I'm interested in, but I've never really felt that I have a ...
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Degree of a real algebraic variety and regular morphisms
I'm reading Fulton's "Intersection theory", which i need for some applied needs.
And i have two questions on general definition of degree used in Fulton.
1)Let us we have a real algebraic variety ...
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Intersection positivity for curves and surfaces
Let $X$ be a smooth complete variety over an algebraically closed field of dimension $\geq3$. Given a divisor $D_1$ on $X$ with $D_1 \cdot C>0$ for every curve $C \subset X$, and a divisor $D_2$ ...
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Dimension of affine variety
Assume that I have $k$ polynomials $f_1(x_1,\ldots x_n),f_2(x_1,\ldots x_n),\ldots f_k(x_1,\ldots x_n)$ in $n>k$ variables. Is it possible to calculate, ,i.e., does there exist a fast algorithm, ...
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if f is birational, is the pushforward map on the numerical groups surjective?
this question was asked on mathunderflow but no one gave a satisfactory answer (perhaps here it will receive more attention?)
Say that one has a morphism of projective algebraic varieties $f: X \to Y$...
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Lower bound for intersection number
The base scheme is an algebraically closed field.
Let $X\to \mathbf{P}^1$ be an arithmetic surface over $\mathbf{P}^1$ and let $P$ be a section of $X\to \mathbf{P}^1$. Let $D$ be an effective (edited)...
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Is there a section disjoint from 0, 1 and infinity on the projective line
Let $K$ be a number field with ring of integers $O_K$. Is there a section of $\mathbf{P}^1_{O_K}$ over $O_K$ whose image is disjoint from $0$, $1$ and $\infty$? If $K=\mathbf{Q}$ this is not possible ...
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What do higher Chow groups mean?
Let $z^i(X, m)$ be the free abelian group generated by all codimension $i$ subvarieties on $X \times \Delta^m$ which intersect all faces $X \times \Delta^j$ properly for all j < m. Then, for each i,...
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Is -(E,E) greater or equal to 2 for a minimal resolution
I'm quite confused by the terminology minimal resolution and minimal model.
Let $f:X\longrightarrow Y$ be a minimal resolution of singularities, where $Y$ is a normal surface.
Let $E$ be an ...
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intersection number
I vaguely recall the following fact that I'd like to use in my research. It should be easy to see that this holds (if it does) but I can't seem to prove it.
Let $p:X\longrightarrow S$ be a (regular) ...
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Curves with negative self intersection in the product of two curves
I wonder if the following is known: Are there two compact curves C1 and C2 of genus>1 defined over complex numbers, such that their product contains infinite number of irreducible curves of negative ...
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Are Chow groups generated by local complete intersections?
Let $X$ be a smooth projective variety over an algebraically closed field. The Chow group $\mathbb Q\mathrm{CH}^d(X)$ is $\mathbb Q$--linearly generated by irreducible subvarieties $Z \subseteq X$ of ...
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intersection theory
Assume that $X$ is a smooth 3-fold (over $\mathbb C$). Let $V$ be a smooth divisor on $X$ and let $S_1,S_2$ be prime divisors on $X$. Assume that given a curve $C$ on $X$ not contained in $V$, then
$...
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A Theorem in Intersection theory.
Fulton's Book on intersection theory (Pg.223, theorem 12.3) asserts the following result:
For r pure dimensional schemes in P^n, whose co-dimensions add to at most n, the product of their degrees is ...
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Survey article on Intersection Theory
Does anybody knows about good overview on intersection theory.
The book of Fulton has very hard language. Does there exist simple overview on this topic with many examples?
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Simple description of a Chow ring of blow-ups.
Is there a simple description of a Chow ring of a blow-up of a point on a smooth projective variety? Or at least of successive blow-ups of $\mathbb{P}^n$?
Maybe something like $A(\tilde{X})=f^*(A(...
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How I calculate degree of the algebraic curve?
Let F be algebraically closed field. Let C be a curve in F^n defined as zeroes of polynomials $p_1(x_1,\ldots,x_n),..,p_{n-1}(x_1,\ldots x_n)$.
Let us define degree of the curve as $\max_S \{ S\cap C ...
- 42.9k
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Chern classes of pushforwards
Let $f:X\to Y$ be a proper morphism of normal varieties (smooth as DM stacks, but I only care about the coarse spaces). The map $f$ is generically finite, but not flat (so no hope of smoothness and ...
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intersection theory on proper algebraic spaces
I have a question about the second example in Hartshorne's Algebraic Geometry, Appendix B, section 3 (given by Hironaka?). It is an example of a compact complex Moishezon 3-fold $X$ which is not an ...
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A composition of a finite morphisms with the transpose correspondence: is it the multiplication by the degree?
Let $X,Y$ be smooth varieties over a field $k$ (which in my case is perfect of finite characteristic $p$; we may also assume that $X,Y$ are connected); $s:Y\to X$ is a finite morphism of degree $d$. ...
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Upper bound on the number of intersections of algebraic manifolds with affine planes
Given an algebraic map $f: B^d \to \mathbb{R}$, from the unit ball of dimension $d$ to the real, let $Y = f^{-1}(0)$. Then it is always possible to find a smaller ball $B_r \subset B^d$ not ...
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Riemann-Roch and Grothendieck duality: general case of Fulton's example 18.3.19
Fulton's "Intersection theory" book contains the following fact (example 18.3.19):
Let $X$ be a Cohen-Macaulay scheme over a field. Assume $X$ can be imbedded in a smooth scheme (so it has a ...
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Chow Ring of Moduli Space of Abelian Varieties
Is there a good reference for the structure of the Chow ring of $\mathcal{A}_g$, the moduli space of complex principally polarized abelian varieties? More generally, references for the intersection ...
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Context for intersection theory
This is a pretty basic question. Hartshorne defines "intersection multiplicity" for any two divisors on a surface. Fulton has an impressive framework of generalizing this in his book (my understanding ...
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On finite endomorphisms of $\mathbf{P}^r$
This question is basically on applying the Grothendieck-Riemann-Roch theorem to finding a formula for the push-forward of a line bundle on $\mathbf{P}^r$ under a certain morphism. Since I have a lot ...
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Family of Enriques surfaces and GRR, Part 2
As I mentioned in my previous post, I am studying the article Moduli of Enriques surfaces and Grothendieck-Riemann-Roch.
The Grothendieck-Riemann-Roch theorem is applied there to show that, for any ...
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