Questions tagged [intersection-theory]

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5
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2answers
432 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
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0answers
45 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
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0answers
148 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
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0answers
105 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 $n-r$ ...
8
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0answers
197 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
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1answer
206 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 ...
5
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0answers
133 views

when does intersection multiplicity jump?

Let $Z$ be a smooth variety and let $X \subset Z \times T, Y \subset Z \times S$ be flat families of closed subvarieties on $Z$, where $T, S$ are smooth and connected, such that for some fixed $z \in ...
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0answers
39 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 ...
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0answers
128 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
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0answers
149 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
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0answers
104 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
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1answer
150 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
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0answers
51 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
3answers
463 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
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1answer
333 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 $\alpha_1$ ...
4
votes
1answer
150 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
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0answers
46 views

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 ...
7
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1answer
368 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, ...
6
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0answers
188 views

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 ...
7
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0answers
115 views

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&...
8
votes
1answer
383 views

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 ...
4
votes
1answer
284 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 ...
2
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0answers
131 views

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 ...
2
votes
1answer
158 views

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 ...
2
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0answers
152 views

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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0answers
139 views

The morphisms induced by two Cartier divisors

Let X be a projective variety. We consider two Cartier divisors $D,E$ such that $E\geq D$ and the relative morphisms $\phi_D: X - - -> \mathbb{P}(H^0(X, O_X(D))^*)$ and $\phi_E: X- - -> \mathbb{...
2
votes
1answer
192 views

Polarization of an abelian variety made by the sum of two divisors

Let $X$ be an abelian variety of dimension $n$, and let $L$ be a polarization, that is, an ample line bundle on $X$, with $\chi(L)=3$. In my specific case, I have that $L=\mathcal{O}_X(\Theta + D)$, ...
3
votes
0answers
214 views

Linear system on singular plane curve

Let $C \subset \mathbb{P}^2_k$ an irreducible plane curve of degree $d >1$ over algebraically closed field $k$. That is $C=V(f(x,y,z))$ where $f \in k[x,y,z]$ homogeneous of degree $d$. Let $\{...
2
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1answer
336 views

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 ...
3
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0answers
107 views

Automorphism group of a hypersurface and its sections

This question is moved from my StackExchange. Assume the base field is algebraically closed. Let $X\subset \mathbb P^n$ be a fixed smooth hypersurface of degree $d$. For any hyperplane $H\subset \...
3
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0answers
155 views

Noether intersection multiplicity for complete intersections

If I take two curves $C,D$ on a surface $M$ with isolated intersection point $p$, then Noether gives a formula equating the intersection multiplicity $i_p(C,D)$ of $C$ and $D$ at $p$ in terms of their ...
1
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0answers
116 views

Formula for fibre square (from Fulton's Intersection Theory)

I have a question about argumentation used in the proof of Proposition 1.7 in Fulton's book on Intersection Theory on page 18: Proposition 1.7 Let $\require{AMScd}$ \begin{CD} X' @>{g'}>> ...
1
vote
1answer
152 views

Schubert cycles that intersect generically transversely

Let $\mathcal{V}= 0 \subset V_1 \subset \cdots \subset V_{n-1}\subset V_n=V$, $\mathcal{W}=0 \subset W_1 \subset \cdots \subset W_{n-1} \subset W_n=W$ be two flags. We say that $\mathcal{V}$ and $\...
4
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0answers
201 views

K-theoretic derivation of Bézout theorem

In the paper "$K$-Theory and Intersection Theory" by Henri Gillet in Handbook of K-theory, 2005 (link behind paywall at Springerlink), the author says: "When the ground field $k = \mathbb C$, Bézout’...
10
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0answers
365 views

The virtual fundamental class as derived intersection

Say $X$ is a smooth projective variety and $\beta\in H_2(X)$ is a class. Then there is a finite-type proper scheme (or in general, stack) $SM : = \overline{\mathcal{M}}_{g,n}(X,\beta)$ of stable maps ...
1
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0answers
54 views

Intersection inside normal cone

For a regular embedding $X\subset Y$, one can think the intersection class $[X]\cdot [X]$ as the intersection of the perturbation of the zero section inside $N_X Y$, intersect with itself. For non-...
-2
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1answer
163 views

Existence of divisor in the Jacobian of smooth curve of genus two whose intersection with theta divisor is 1

Let $C$ be a smooth projective curve of genus $2$ and $J$ denotes the Jacobian of $C$. Let $\theta$ be the image of $C$ under the abel Jacobi map. Is there exist a divisor $D$ in $J$ such that $D.\...
2
votes
1answer
267 views

Join of two intersecting varieties

Suppose I have two smooth projective varieties $X$ and $Y$ in $\mathbb{P}^n$, that intersect along a smooth subvariety $Z$. Is there a formula to compute the degree of the join variety $J(X,Y)$ of $X$ ...
1
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0answers
86 views

Factorizations of closed embeddings of smooth schemes

All schemes will be of finite type over a field $k$. Say I have a closed embedding $\iota: X \hookrightarrow Y$ of smooth $S$-schemes for some scheme $S$ (in particular it is a regular embedding). ...
1
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0answers
80 views

Numerical and rational equivalences on intersection of divisors

Let $X$ be a smooth projective variety over a finite field. Since $Pic^0(X)$ is finite and $Pic^0(X)$ can be identified with numerically equivalent to zero divisors this implies that for divisors on $...
2
votes
1answer
212 views

Exact sequence of normal cones

Suppose that we have a sequence $i: X \hookrightarrow Y$, $j: Y \hookrightarrow Z$ of closed embeddings of varieties such that $i$ is regular. In this case, do we have an exact sequence of cones of ...
1
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0answers
133 views

Log-canonical bundle of a smooth curve with marked points

I am not sure if this question is appropriate for this site, but here it goes. I am not a geometer, so I am not familiar with notation in the area. I am interested in the moduli space of $r$-spin ...
6
votes
1answer
270 views

Severi Formula for Intersection Multiplicities

I say in advance that I am a novice in Intersection Theory, so forgive me if my question is trivial. Let $X\subseteq\mathbb{P}^N$ be a smooth irreducible projective variety of dimension $n$ and $V, W\...
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0answers
167 views

Topological vs algebraic intersection forms

Let $X$ be a simply connected complex projective surface (hence a real $4$-manifold). Let $(H^2(X,\mathbb Z)/\mathrm{tors}, q_X)$, $(A^1(X)/\mathrm{tors},q_X')$ be the corresponding lattices in ...
2
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0answers
144 views

Surjectivity of pushforward on Chow rings for stacks

Let $f:X\rightarrow Y$ be a proper morphism of smooth Deligne-Mumford stacks of finite type over $\mathbb{C}$ that is birational, but not flat. The coarse spaces of $X$ and $Y$ are both not smooth. Is ...
1
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0answers
287 views

Functoriality of Chern-Fulton's class

Let $X$ be a, possibly singular, algebraic variety embedded as a closed subvariety of a manifold $M$ with map $i : X \rightarrow M$, and $\pi : \tilde{M} \rightarrow M$ be a proper birational map with ...
1
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0answers
70 views

Is it possible to represent a closed substack as a fundamental cycle?

Let $X$ be an Artin stack and $Z \subset X$ be a closed substack. Can we represent $Z$ as a fundamental cycle? i.e. $[Z] = \sum_i a_i [Z_i]$ where $Z_i$ are integral substacks of $X$. In other word, ...
3
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0answers
77 views

Quartic link in a 5-sphere

In this post I would like to propose a quartic link in a 5-sphere. Let us start with the following gluing into a 5-sphere: $$S^5=(D^2_{} \times T^3_{}) \cup_{T^4} ({S^5 \smallsetminus D^2 \times T^3})...
2
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0answers
166 views

Triple link in a 5-sphere — Proposal

In this post I would like to propose a triple link in a 5-sphere. Let us start with the following gluing into a 5-sphere: $$S^5=(D^2_{} \times T^3_{}) \cup_{T^4} ({S^5 \smallsetminus D^2 \times T^3})$...
2
votes
1answer
189 views

Vector bundles on henselian schemes

Let $X$ be a smooth and projective scheme over $\mathbf{Z}_p$. We call $\mathfrak{X}$ the ringed space whose topological space is the topological space of the special fiber of $X$, and whose ...

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