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# 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)

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### Intersection of divisor and curve on a subvariety

Let $X$ be a normal $\mathbb Q$-factorial variety (irreducible) over an algebraically closed field $k$ of characteristic $0$. Let $D\subseteq X$ be an irreducible divisor (which must be $\mathbb Q$-...
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### Fundamental cycle

Let $X$ be a seperated scheme of finite type over a field. We define the fundamental cycle of an equidimensional subscheme $Z \subset X$ with irreducible components $Z_1, ..., Z_r$ to be the $d$-cycle ...
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### Is there a relatively easy way to find whether two plane curves have a "common component"?

I am interested in determining all intersections between two plane curves f(x,y)=0 and g(x,y)=0. (f is degree 4, and g is degree 3.) I would like to use Bezout's Theorem. I have found 12 total ...
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This answer gives the first three Chern classes of the tensor product of any two locally-free sheaves. I computed the fourth Chern class as $$c_4(E\otimes F) = \frac{1}{2} c_1^4(E) -5c_1^3(E)c_1(F) -\... 0 votes 0 answers 53 views ### The Self-Intersection Number of the Complex Projective Line \mathbb{C}\mathbb{P}^1 Let \mathbb{C}\mathbb{P}^2 be the Complex Projective Plane. Let C \subset \mathbb{C}\mathbb{P}^2 be a Rational Curve, i.e., C is isomorphic to the Complex Projective Line \mathbb{C}\mathbb{P}^1... 0 votes 0 answers 17 views ### Confusion about why a specifc Vector Transformation magically fixes the UV mapping of a Rotating ray plane intersection Alright so, i have been toying around with some basic ray intersection functions as of recently. And in doing so found myself rotating these intersection shapes, such as disks. Which eventually lead ... 0 votes 0 answers 37 views ### Why doesn't GeoGebra show the points of intersection between two ellipses when they are tangent? I drew two ellipses in GeoGebra that have a common focus and my purpose is to show all cases of their relative geometries. I used the Intersect(Object,Object) ... 5 votes 1 answer 200 views ### Blow-up of a Pencil of Cubic Curves (Miranda's basic theory of elliptic surfaces) In Rick Miranda's "The basic theory of elliptic surfaces" the Example (I.5.1) see page 7 on a pencil of plane curves contains an argument Inot understand yet: Let C_1 be a smooth cubic ... 0 votes 1 answer 33 views ### Compute the intersection form Consider a smooth subvariety \iota:X=V_+(f)\subset\mathbb{P}^2\times\mathbb{P}^1 with some f\in H^0(\mathcal{O}(1,2)), how can I compute the intersection$$\iota^*\mathcal{O}(2,3).\iota^*\mathcal{... 