# Questions tagged [intersection-theory]

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Consider homogeneous polynomials $P_0,P_1,P_2,P_3,P_4,P_5$ of degrees $3,3,2,3,2,1$ over $\mathbb{P}^3$, and the map $\phi:\mathbb{P}^3\rightarrow\mathbb{P} = \mathbb{P}(3,3,2,3,2,1)$ given by $$\phi(... 3 votes 2 answers 172 views ### Moving lemma for countable collection of subvarieties Fix an integer n \ge 5. Let \mathcal{V} be a countable collection of closed subvarieties of \mathbb{P}^n_{\mathbb{C}} of codimension at least 2. Choose a point p \in \mathbb{P}^n. Does there ... 3 votes 1 answer 147 views ### Extending effective Cartier divisors Let X be a non-singular, quasi-projective variety (over \mathbb{C}) of dimension at least 3, D_1, D_2 are integral effective divisors in X with D_1 \cap D_2 of codimension 2 in X. Let ... 2 votes 0 answers 92 views ### On intersections of exceptional divisors Let X be a smooth, projective variety of dimension n \ge 2, L a very ample line bundle on X and \pi: \widetilde{X} \to X be the blow-up along a closed subvariety of codimension at least 2. ... 4 votes 2 answers 211 views ### Is the sum of a radical ideal and the ideal of a generic linear space intersecting that ideal radical? Let X \subseteq \mathbb{C}^n be an irreducible algebraic set that forms a cone, and let I=I(X) \subseteq \mathbb{C}[x_1,...,x_n]. Let m < n and k\leq m be positive integers. Is it true that ... 1 vote 0 answers 54 views ### Use of Porteus‘ formula in a paper of Beauville In “Sur la cohomologie de certains espaces de modules de fibrés vectoriels”, Beauville calculates the Chern class of the diagonal \Delta of the moduli space M of certain stable bundles on a curve ... 3 votes 0 answers 62 views ### Detecting non-principal Weil divisors on normal varieties using curves Let X be a normal projective variety over an algebraically closed field k. Given any morphism f:Y\to X, there is a pullback homomorphism f^*:\text{Cl}(X)\to\text{Cl}(Y), where \text{Cl}(X) ... 1 vote 0 answers 129 views ### Pinch points and dual surfaces I am currently reading Fulton's expository lectures "Introduction to intersection theory in algebraic geometry". On pg. 4, Fulton sketches an argument of George Salmon which I don't ... 4 votes 1 answer 242 views ### Intersection of curves in non-singular projective algebraic surfaces Bezout thereom that says that two irreducible algebraic curves C and D in \mathbb{P}^2_\mathbb{C} intersect at nm points (counted with multiplicity), where n and m are the degrees of C ... 2 votes 1 answer 155 views ### Are "transverse" hyperplane sections of nondegenerate irreducible projectice varieties always nondegenerate Let X \subseteq \mathbb{P}^n be a irreducible complex projective variety. It is called nondegenerate if it is not contained in a hyperplane in \mathbb{P}^n. Assuming X is nondegenerate and ... 5 votes 1 answer 256 views ### Intersection cycle in a product of Grassmannians Let G(k,n) denote the Grassmiannian of k-planes in \mathbb C^n. Let's define$$ I_j =\{ (\Lambda_1,\Lambda_2 ) \in G(k,n) \times G(l,n) \, | \, \dim(\Lambda_1 \cap \Lambda_2) \geq j \}. $$These ... 2 votes 1 answer 86 views ### Cycle of non-equidimensional scheme In Fulton's intersection theory, example 1.7.1, he mentioned an example that contradicts to the splitting of cycles with respect to irreducible components. Consider the subscheme X in \mathbb{A}^3 ... 5 votes 1 answer 266 views ### Algebraic K-theory and intersection theory (Bloch's formula) It seems to be a well known fact that algebraic K-theory can be used to understand intersection theory, at least for varieties (or stacks!) over a field. A first glimpse of this result seems to be ... 3 votes 0 answers 84 views ### A proper morphism restricts to a closure of a point on the generic fiber Let \pi:X^{e}\rightarrow Y^{d}(e\geq d) be a proper and dominant morphism of projective varieties over field k. Moreover, Y is assumed to be smooth. Denote \eta the generic point of Y, X_\... 1 vote 0 answers 73 views ### Varieties swept out by Linear Spaces nondegenerated We working over complex numbers \mathbb{C} keeping our constructions as geometric as possible. Let \Lambda_1, ..., \Lambda_m \cong \mathbb{P}^{n-2} \subset \mathbb{P}^{n}  be pairwise distinct, ... 6 votes 1 answer 288 views ### Computing Massey products via intersection theory Let K be an n-manifold with boundary and let x,y,z \in H^*(K) be cohomology classes with x\cup y=y\cup z=0. The Massey product \langle x,y,z \rangle is defined as the set of cohomology ... 1 vote 0 answers 147 views ### Genus of a curve given by self intersection of a very ample line bundle Let X be a smooth, integral and projective d-dimensional variety over a field k of characteristic 0. Let H be a very ample line bundle over X. Assume that there exists a smooth and ... 1 vote 1 answer 98 views ### Line segment-triangle intersection algorithm [closed] currently in my project I'm using signed tetrahedron volume to check whether a line segment intersects a triangle. Initially I've found this approach in the great answer provided by professor O'Rourke:... 2 votes 1 answer 110 views ### Curves sharing points over finite fields, and their mutual divisibility Consider in \mathbb{A}^2(\mathbb{F}_q) two \mathbb{F}_q-rational curves \mathcal{X}:f(x,y)=0 and \mathcal{Y}:g(x,y)=0, and let \mathcal{Y} be absolutely irreducible. Suppose also that \... 6 votes 0 answers 212 views ### Singling out irreducible components Let V\subset \mathbb{A}^n be a variety defined by equations of degree \leq D, or, what is the same, an intersection of hypersurfaces of degree \leq D. Let V^+_0 be an irreducible component of ... 3 votes 0 answers 93 views ### Singling out lower-dimensional components Let V\subset \mathbb{A}^n be defined by equations of degree \leq D. (That is, V is an intersection of hypersurfaces of degree \leq D.) Assume V is not pure-dimensional. Let V^- be the ... 0 votes 0 answers 103 views ### Intersection product when one factor is contained in the exceptional divisor I am trying to calculate some intersection numbers and would appreciate help on the following problem: Consider two divisors D_1 and D_2. Blowing up their intersection yields \varphi^{*}(D_i) = \... 5 votes 2 answers 543 views ### Reference request: Kleiman's proof of Snapper's Lemma On page 4 of Nitin Nitsure's paper Construction of Hilbert and Quot Schemes, the author refers to the fact that Hilbert polynomials are indeed polynomials as a special case of Snapper's Lemma, see &... 2 votes 0 answers 63 views ### Comparing the Segre classes of a cone with its abelian hull Let X be a smooth scheme, with a sheaf of graded quasi-coherent algebras \mathcal{A}^*, that yields a cone C (in the sense of Fulton's intersection theory). Suppose that \mathcal{A}^1 is a ... 3 votes 0 answers 165 views ### algebraic vs rational equivalence Are there classes of algebraic varieties for which algebraic and rational equivalence for algebraic cycles coincide? (references also appreciated) 3 votes 0 answers 139 views ### Transversal intersection with linear subspaces Let us work over an algebraically closed field K. If X\subset \mathbb{P}^n is a closed subset of dimension r, then there should exist a linear subspace L\subset \mathbb{P}^n of dimension n-r ... 9 votes 0 answers 246 views ### Integrality of primary genus 0 Gromov-Witten invariants of a Fano manifold Suppose (X,\omega) is a positively monotone compact symplectic manifold, i.e., after a positive scaling of the symplectic form, we have c_1(T_X) = [\omega] in de Rham cohomology (T_X has well-... 3 votes 1 answer 351 views ### Intersection theory on singular varieties by embedding to smooth ones Let X be a normal complex projective variety over \mathbb C. In order to define the intersection product of the Chow ring, one usually requires X to be smooth. How to weak the smoooth assumption ... 0 votes 0 answers 47 views ### EXACT number of intersection points of two algebraic curves As the picture shows2(the paper's link is in 1),it seems that I can use tools including Bezout's theorem to solve the EXACT number of intersection between two algebraic curves(F(x,y) is of degree two ... 1 vote 0 answers 142 views ### How to compute the genus of the (singular) intersection of three quadratics in \mathbb{C}P^4? Consider three quadratics in \mathbb{C}P^4:$$ x_0^2+4x_1^2+\frac{x^2_2}{4}=0,~ x_1x_4+x_2x_3=0, ~ x_0^2+x_3^2+x_4^2=0. $$If there intersection was non-singular, then the intersection should be a ... 5 votes 0 answers 158 views ### How to compute the class defined by intersection with a square? \DeclareMathOperator\Gr{Gr}Let \Gr(k,n+k) (of course, one can do also for \Gr(k,\infty)) be the complex Grassmannian of k-planes in n+k-dimensional linear space. It is well-known that ... 2 votes 0 answers 138 views ### Rank of the top Chow group Let X be a regular integal scheme of finite type over \mathbb Z and assume that X has dimension d. In general it is not known if the Chow groups CH^q(X) (q is the codimension) are finitely ... 1 vote 1 answer 256 views ### Non-transverse intersection of submanifolds What can we tell about non-transverse intersection points of (smooth) submanifolds? Especially, in the case of complementary dimensions, how can we define and calculate the ''multiplicity'' of an ... 1 vote 0 answers 52 views ### Locus of linear spaces with prescribed contact order Let X\subset\mathbb{P}^{n} be a smooth projective variety of pure dimension d. Let Z\subset \mathbb{G}(n-d,n)\times\mathbb{P}^{n} be the space of pairs (P,x) of a linear space P\cong\mathbb{P}... 4 votes 3 answers 481 views ### Irreducible components: associativity for intersections? Let A, B, C be closed irreducible subvarieties of \mathbb{A}^n. Let V_1 be an irreducible component of B\cap C, and V an irreducible component of A\cap V_1. Must there necessarily be ... 5 votes 1 answer 361 views ### Statements related to Thurston's work on the surface If we have simple closed curves \alpha and \beta on a surface \Sigma_g, the intersection number i(\alpha ,\beta) is defined to be the minimal cardinality of \alpha_1\cap\beta_1 as \alpha_1 ... 1 vote 1 answer 330 views ### Push-forward of divisors and intersections Let f:X\rightarrow Y be a surjective finite morphism of varieties, with X normal and Y smooth. Let D\subset X be a divisor and C\subset Y a curve. Does the equality$$C\cdot f_{*}D = f^{*}C\... 1 vote
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### Minimising kurtosis (non-convex). Can I use algebraic geometry or alternate methods to show uniqueness of a particular solution?

I consider a weighted sum of $n$ identically-distributed correlated random variables. The weights in the sum, $w_i$ for $i=1, 2,...,n$, satisfy $w_i>=0$ and $\sum_{i=1}^{n}w_i=1$. I am ...
435 views

### Intersection theory in analytic geometry

This might be a weird/stupid question, but it came to me a couple of times, and I would like to get an answer for that. In some papers I read, constantly the authors define some analytic subspaces, ...
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### If cohomology theory corresponds to intersection theory, valuation theory corresponds to -?

This is a meta question I asked myself. Cohomology theory is dual to an intersection theory. Is there anything valuation theory corresponds to in general? For instance, McMullen's polytope algebra is ...
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### Intersection numbers via residue formula

$\newcommand{\sslash}{\mathbin{/\mkern-7mu/}}$With a friend we are trying to understand residue formulas in the article "Cohomology pairings on singular quotients in geometric invariant theory&...
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### Interpretation of "27" lines for cubic surface with rational double points

It is well known that a smooth cubic surface has $27$ distinct lines. Explicitly, if we choose a planar representation, i.e., blowup $\mathbb P^2$ at $6$ general points $p_1,...,p_6$, the $27$ lines ...
327 views

### Relative canonical class of blowing-up a flag ideal

Let $X$ be a smooth complex projective variety of dimension $n$. Consider a flag ideal $I$ on $X\times \mathbb{P}^1$, namely, $$I=I_0+I_1t+I_2t^2+\cdots+I_{N-1}t^{N-1}+(t^N)\,,$$ where $t$ is the ...
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### Localization of Chow groups and flat base change

For any flat morphism $f:X\rightarrow Y$, we have a flat pullback of Chow groups $$Ch^i(Y)\rightarrow Ch^i(X).$$ A particular example of this is of course an open immersion $U\rightarrow X$. In that ...
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### Upper semi-continuity of intersection numbers

Consider a smooth projective morphism of schemes $X \rightarrow S$ with relative dimension $n$ (the application I have in mind is with $S$ = an open subset of $\text{Spec } \mathbb{Z}$) and assume ...
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### Analytic and algebraic definitions of intersection multiplicity of two complex algebraic curves coincide

There are two definitions of intersection multiplicity of two complex algebraic plane curves. One of them is given in An introduction to algebraic curves by Griffiths. Let $t \mapsto (t^k, y(t) )$ be ...
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### Looking for examples of not injective maps and not surjective maps of the form $A_{k} (X) \to H_{2k} ( X , \mathbb{Z} )$
Here: https://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/alggeom-2002-c9.pdf, on pages: $1$ and $2$, we find the following paragraph: For any scheme of finite type over a ground field ...