# Questions tagged [intersection-theory]

In mathematics, intersection theory is a branch of algebraic geometry, where subvarieties are intersected on an algebraic variety, and of algebraic topology, where intersections are computed within the cohomology ring. (Ref: http://en.m.wikipedia.org/wiki/Intersection_theory). Do not use this tag for elementary problems in linear algebra or geometry. (e.g. determining whether two lines intersect)

388 questions
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### plane edge intersections embedded in higher dimensional space

Let's say we have some D-dimensional euclidean space, and we have some circles of dimension 0 to D-1 (circle dimensionality meaning the minimum number of vectors needed to fully define it, so a ...
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### Codimension 1 points

Today I was reading a proof of the following Lemma from Liu's "Algebraic Geometry and Arithmetic Curves" Recall: A a point $x \in X$ is called a codimension 1 point if $\overline{ \{x \}}$ has ...
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### Transversality in the Zariski tangent space

I am trying to understand how to check whether two algebraic varieties intersect transversally from a purely algebraic standpoint. Is the following argument correct? Say locally a smooth projective ...
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### Chow ring isomorphic to homology ring?

I heard an algebraic geometry professor say that the Chow ring is usually isomorphic to the homology ring for cases we care about in application. However, I cannot find many results about this, beyond ...
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### Polygon and line intersection

Does anyone help me with the fast algorithm to determine the intersection of a polygon (rotated rectangle) and a line (definite by 2 points)? The only true/false result is needed.
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### singular intersection only comes from tangent?

Let $X$ be a smooth hypersurface in $\mathbb {CP}^n$, $H$ be a hyperplane. If $X\cap H$ is singular, is it true that $H$ is a tangent plane at some point of $X$? Note the converse is always true, if ...
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### “intersection number” of surface with boundary

Consider in an $2n$ dimensional manifold $M$(compact, smooth), for embedded $n$ dimensional surfaces with boundary, denoted by $\Sigma$ and $\Sigma'$, in M, we can consider "intersection number"(...
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### Intersection number and cup product

It is known that for a closed oriented smooth manifold $M$, if A and B are oriented submanifolds of $M$, and if A and B intersect transversely, then the Poincare dual of A ∩ B is the cup product of ...
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### Intersection of Schubert cycles

I want to compute intersection of the Schubert cell $\sigma_{(3,0)}$ with all the cells $\sigma_{a_1, a_2}$ in the grassmanian $G(2,5)$. I am not sure I am doing correctly but I can't see my mistake. ...
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### General statement about how many lines in Euclidean space will determine a line

It is easy to see that in $3$-dimensional Euclidean space, given $4$ lines in general position, there exists precisely one line who intersects with each of the $4$ lines. We call the $4$ lines ...
### Question about adjunction formula $K_Y = (K_X + Y)|_Y$
I've learnt that if $Y\subset X$ is a smooth subvariety of codimension 1 then the degree of canonical divisors of $Y$ and $X$ are related by \deg K_Y = \deg((K_X + Y)\cap Y). \end{...