Questions tagged [intersection-theory]

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1answer
110 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 ...
3
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85 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 curves. One is given in An introduction to algebraic curves by Griffiths. Let $t \mapsto (t^k, y(t^k) )$ be the local ...
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129 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
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1answer
172 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
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0answers
189 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
293 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 ...
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130 views

Degree of varieties swept out by linear spaces (Eisenbud & Harris' 3264 and All That)

I have a couple of questions on statements from Harris' and Eisenbud's lecture "3264 and All That" at page 145, Section 4.2.3: Varieties swept out by linear spaces. The contents of 4.2.3 answers ...
3
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0answers
102 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 \...
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142 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 ...
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108 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
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1answer
133 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 $\...
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192 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’...
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323 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 ...
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52 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-...
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1answer
154 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.\...
1
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1answer
161 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 $X$ ...
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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). ...
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69 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 $...
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78 views

Defining pull-back of Chow groups under a morphism of special type

Let $X$ be a normal complex projective variety (not necessarily smooth), and let $Y$ be a smooth complex projective variety. Let $Z\subset X$ be a smooth closed subvariety. Let $\pi : Y\rightarrow X$...
2
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1answer
191 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 ...
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108 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
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1answer
261 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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166 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
124 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 ...
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201 views

Blow up example

I am working in this case: Let X be a, possibly singular, algebraic variety embedded as a closed subvariety of a manifold(smooth) M$(i : X \longrightarrow M )$. We consider the Blow up $\pi$ of $M$ ...
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66 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, ...
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0answers
76 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})...
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0answers
165 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
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1answer
180 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 ...
2
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0answers
156 views

Splittings in the difference bundle construction of Atiyah-Hirzebruch

I'm reading the construction of difference bundle(generalized) in the paper of Atiyah-Hirzebruch(Analytic cycles on complex manifolds) There is one thing I cannot understand. The followings are in ...
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0answers
91 views

Equivalence relations among Algebraic Cycles

In the book 3264 and All That-Eisenbud & Harris, the authors claim that for smooth projective varieties admitting an affine stratification, the algebraic equivalence relation and the rational ...
2
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1answer
165 views

On connectedness of intersection of subgroups

I am quite interested in any partical answer to the following general (maybe a little bit vague) question: Is there some criterion about the connectedness of the intersection of two connected ...
1
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0answers
159 views

Continuity of Intersection Pairing on Chow monoids

Let $X$ be a smooth irreducible complex projective variety. As we know, if $\alpha,\beta$ are two cycles intersecting properly in $X$, we can define, via Serre's Intersection Formula, their ...
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0answers
37 views

Counting Zeros Under Unitary Action

Assume we have two polynomials $f_1$ and $f_2$ with Newton polytopes $A_1 , A_2 \in \mathbb{Z}^2$. Also suppose that coefficients of $f_1$ and $f_2$ are generic. Then we pick a unitary matrix $Q$ and ...
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0answers
111 views

What happens to a variety after a change of variables?

Suppose I have an irreducible affine variety $X \subseteq \mathbb{A}^n_k$. Let us denote $X = \{ x \in k^n : f_j(x) = 0 \ (1 \le j \le M) \}$. $k$ is an algebraically closed field. Let $a_i \in k$, $...
2
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1answer
296 views

A “boundary map” for the algebraic equivalence relation of cycles

In what follows by projective variety I will mean the zero locus of homegeneous polynomials in some projective space. Anyway, feel free to deal with a sufficiently good scheme if you want. Let $X$ be ...
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0answers
346 views

Applications of E8 manifold

The $E_8$ Cartan matrix is given by, $$ K_{E_8}=\begin{pmatrix} 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 2 & -1& 0 & 0 & 0 & 0 & 0 \\ ...
3
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1answer
170 views

Homotopy of paths at the boundary

Denote by $\Gamma$ a hypersurface in $\mathbb{C}^2$, i.e. the zero locus of a polynomial of two complex variables. Denote by $X$ the complement of $\Gamma$ in $\mathbb{C}^2$. I am trying to define a ...
4
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1answer
213 views

Pairing on arithmetic surfaces

Let $f: X \to S$ be an arithmetic surface, where $S=\operatorname{Spec } O_K$ for a number field $K$. It is well known that if we want to introduce a reasonable intersection theory on $X$ we have to ...
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0answers
42 views

Hypersurface whose “square” level sets intersect all linear subspace of “high” dimension

Let $k$ be an infinite perfect field (e.g. I'm happy to assume that $k$ has characteristic $0$. On the other hand, the algebraically closed case is not interesting for this question). The question is ...
4
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1answer
156 views

Proper locally trivial bundle is injective on Chow groups

If $X\to Y$ is a map of varieties that is Zariski-locally isomorphic to a projection $U\times P\to U$ with $P$ (smooth) proper, I think the pullback $A_{\bullet}(Y)\to A_{\bullet}(X)$ is supposed to ...
5
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1answer
349 views

Fulton's deformation to the normal cone vs Verdier's

Let $X$ be a smooth variety over a field $k$, and let $Y$ be a smooth subvariety. In the literature, I've seen two versions of the deformation to the normal cone: Verdier's version: $\tilde{X}_Y^\...
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0answers
198 views

Bezout theorem for germs of holomorphic functions

UPDATE. It was pointed out by @Dmitri that two smooth curves given by $f=y$ and $g=y+x^k$ in $\mathbb C^2$ provide a simple counterexample. Let $f_1, \ldots, f_p, g_1, \ldots, g_q$ be germs of ...
6
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0answers
209 views

Schubert cycles on Grassmannian bundles

Let $X$ be smooth variety and let $\mathcal{E}$ be a vector bundle on $X$ of rank $n$. On the total space of the Grassmannian bundle $\pi:G(k,\mathcal{E})\to X$ we have the tautological exact sequence ...
3
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0answers
218 views

The Chow ring of a blow-up along a badly embedded subscheme

Let $X$ be a variety and $i:Y\hookrightarrow X$ be a regularly embedded closed subvariety, and write $\tilde X$ for $Bl_Y X$ the blow-up of $X$ along $Y$. Let $A^*(X)$ refer to the Fulton-Macpherson ...
3
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1answer
331 views

Intersections with a Power of an Ample Divisor on an Abelian Variety

Let $A$ be a $g$-dimensional, complex abelian variety, let $H$ be an ample divisor, let $D\in Pic^0(A)$, and let $0\leq k\leq g$. Question 1: Does $D^k\neq 0\in CH^k(A,\mathbb{Q})$ imply that $H^{g-k}...
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87 views

$ch(L f^*\epsilon)$

I know the famous principle (or theorem depending on how you define) of Chern classes for locally free sheaves and a morphism $f: X'\rightarrow X$, $ch(f^* \epsilon)=f^* ch(\epsilon)$. But if $f$ ...
4
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2answers
187 views

Cycle class of zeroes of a global section

Let $\mathcal{F}$ be a locally free sheaf of rank $n$ on an $n$ dimensional complex manifold $X$. If the zero locus of a generic global section of $\mathcal{F}$ is $0$ dimensional, then its cycle ...
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285 views

Twisted sheaves on tower of $\mathbb{P}^n$

Take the projective space $\mathbb{P}^n$ over a ring $W$. We call $\mathcal{O}(q)$ the usual twisted line bundle. Now take the map $f: \mathbb{P}^n\to\mathbb{P}^n$ defined by $$[x_0,\ldots, x_n]\...
3
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0answers
268 views

Non algebraizable formal abelian schemes

I'd like to collect a good number of examples of formal schemes over $\text{Spf}(\mathbf{Z}_p)$, whose special fiber is a projective variety over $\mathbf{F}_p$, but that are not algebraizable. If ...

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