Questions tagged [intersection-theory]
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394 questions
5 votes
1 answer
207 views
Different results for a computation on cohomology of Grassmannian
I try to study a specific operation of pullback and pushforward related to flag varieties. Let $Y=Gr(r-1,2r-1)$ be the variety of $r-1$ subspaces in a vector space of dimension $2r-1$. I also have $Y'=...
- 367
0 votes
0 answers
130 views
Tate/semisimplicity conjecture for resolution of nodal quartic surface
Thanks to work of many people (Madapusi, Charles, Maulik...) the Tate conjecture is now known for K3 surfaces over any finitely generated field. A smooth quartic surface is K3. Consider instead a ...
- 1,106
7 votes
1 answer
450 views
Geometric meaning of a resultant
I am working with a particular multivariate resultant and I suspect that behind there is a geometric interpretation that I cannot see, so I would be happy if some one could help me to understand that. ...
- 83
3 votes
1 answer
242 views
Intersection of curves after a blow-up
Let $C,D\subset \mathbb{P}^n$ be irreducible and reduced curves, for some $n\ge 2$. For a point $p\in \mathbb{P}^n$ we can define the intersection multiplicity $$I_{p}(C,D)=\mathrm{dim}(\mathcal{O}_{p,...
- 7,916
8 votes
0 answers
230 views
If the motive of an abelian variety is defined over a subfield, is the abelian variety defined over that field?
Let $L / k$ be a finite extension of fields, let $M$ be a pure motive over $k$ (with rational coefficients), and suppose that the base change $M_L$ of $M$ to $L$ is isomorphic to $\mathfrak{h}^1(A)$. ...
- 1,106
4 votes
0 answers
258 views
$L$-function form of Tate conjecture for divisors on abelian varieties
Let $k$ be the function field of a $d$-dimensional regular integral finite-type scheme $Y$ over $\mathbb{Z}$. Conjecture 2 in Tate's paper in the Woods Hole proceedings predicts (among other things) ...
- 1,106
0 votes
0 answers
78 views
Does the semigroup of covolume polynomials have the cancellation property?
Let $V_n^d(\mathbb{Q})$ denote the set of realizable volume polynomials of degree $d$ in $n$ variables over $\mathbb{Q}$ - these are polynomials of the form $\frac{1}{d!}\int_Y (\sum x_i D_i)^d$ where ...
1 vote
0 answers
187 views
Is there such a projection formula?
The background of my question is the intersection theories introduced in Chapters 42 and 43 of the Stacks Project. We work over an algebraically closed field. Let $Y$ be a smooth projective variety ...
- 265
0 votes
0 answers
153 views
The Chow ring with rational coefficients is generated multiplicatively by Picard group?
I see Kiritchenko's Intersection theory note claims that the Chow ring with rational coefficients of a smooth scheme $X$ is generated multiplicatively by Picard group (page 18). Is this claim true ? ...
- 69
4 votes
0 answers
110 views
Degree of bisector curve of two plane algebraic curves
Assume $C_1$ and $C_2$ are two plane algebraic curves in $E = \mathbb{A}^2$, the affine plane, given by equations $f(x,y) = 0$ and $g(x,y) = 0$ of degrees $d$ and $e$ respectively. Now the bisector ...
- 748
0 votes
0 answers
161 views
Why the need for finite dimensional approximation for equivariant intersection theory?
I have been trying to get into equivariant cohomology from an algebraic geometry perspective, and it seemed like Fulton's most recent book would be a halfway decent place to start. His approach is to ...
- 595
1 vote
0 answers
208 views
How the étale-Chern class compatible with intersection theorical Chern class
Let $X$ be a (nice enough) variety over a field $k$, with a regular closed immersion $i:Z\rightarrow X$. All the cohomologies appeared are supposed to be étale cohomology. Fix the coefficient ring $\...
- 395
3 votes
0 answers
112 views
Why is the Segre class of a cone $C$ equal to the Segre class of the cone $C\oplus 1$?
This question regards example 4.1.1 from Fulton's Intersection theory. Namely we have the following statement: Example 4.1.1. For any cone $C$, $s(C\oplus 1) = s(C)$. Here we define a cone $C$ over ...
- 39
1 vote
1 answer
247 views
Niveau of the Hodge structure of an hypersurface in $\mathbb{P}^n$
Assume $X:=X_d$ is an hypersurface of degree $d$ in $\mathbb{P}^n$. Assume in addition that $d<n+1$, hence $X$ is a Fano variety. The hyperplane section theorem of Lefschetz ensures that the only ...
- 367
1 vote
0 answers
232 views
Canonical divisor and blow-ups
The starting point of my question is the following exercise from Liu's book in the framework of arithmetic surfaces (mixed characteristic). In particular in the following image $S$ is a Dedekind ...
- 159
1 vote
0 answers
173 views
Relative adjunction formula for singular surfaces
The following is the adjunction formula for regular arithmetic surfaces (taken from Liu's book): where $S$ is a Dedekind scheme of dimension $1$ and $X$ is an integral, projective scheme of dimension ...
- 159
2 votes
0 answers
332 views
Intersection signature on divisors of Calabi-Yau threefolds
After over one week and quite a lot of views on this question, I would like to ask a refined version here. Let X be a minimal Calabi-Yau threefold in the sense of [1] and let $D$ be a Weil divisor on $...
- 21
1 vote
0 answers
167 views
Projective varieties with non-isomorphic Chow motive but isomorphic rational Chow rings
I would like to have examples of smooth projective varieties $X$ and $Y$ that have the same rational Chow ring but non-isomorphic Chow motives $h(X)$ and $h(Y)$.
- 3,747
1 vote
0 answers
111 views
Pullback of the canonical divisor along blowup of a singular surface and intersection numbers
Given $X$ a normal projective surface over $\mathbb{C}$ and $p$ be a singular point of $X$ . Let $\pi: Y \rightarrow X$ be the blowup of $X$ at the point $p$. Is it true that $\pi^{\star}(K_{X}) \...
- 75
3 votes
0 answers
234 views
Are motives of K3 surfaces of abelian type?
I refer to the article of van Geemen https://arxiv.org/pdf/math/9903146. What van Geemen calls the Kuga-Satake-Hodge conjecture suggests that for a K3 surface $X$ over $\mathbb{C}$, the summand $h^2(X)...
- 1,106
3 votes
0 answers
201 views
Tate conjecture for singular varieties in terms of intersection homology
In his book “Mixed motives and algebraic K-theory”, Jannsen generalizes the Tate conjecture to a potentially singular projective variety $X$ over a finitely generated field. The statement is the same ...
- 1,106
4 votes
0 answers
142 views
Euler factors from bad primes and the Beilinson-Bloch vanishing conjecture
The vanishing part of the Beilinson-Bloch conjecture asserts that for a smooth projective variety $X$ over a number field $K$, $\dim_{\mathbb{Q}} \operatorname{CH}^i(X) \otimes_{\mathbb{Z}} \mathbb{Q} ...
- 571
4 votes
1 answer
378 views
Known cases of Tate conjecture for varieties which are smooth over a curve
What are some examples of smooth projective varieties $X$ over a finite field for which the Tate conjecture for divisors is known, and which admit a smooth morphism to a smooth projective curve? I am ...
- 1,106
1 vote
0 answers
131 views
Computing Chow groups of affine, simplicial toric varieties
Let $k$ be an algebraically closed field. Let $X$ be an $n$-dimensional affine, simplicial toric variety over $k$. There exists an $n$-dimensional simplicial cone $\sigma$ in $\mathbb{R}^n$ such that $...
- 721
1 vote
1 answer
173 views
Why is $2A_0(X)=0$ for a cubic threefold $X$ containing a line, over an arbitrary field $k$
I can't quite follow Proposition $2.1$ of "UNIVERSAL UNRAMIFIED COHOMOLOGY OF CUBIC FOURFOLDS CONTAINING A PLANE". I posted this on Math stackexchange but got no answer. Let $X$ be a smooth ...
- 729
2 votes
1 answer
323 views
On intersection theory on toric varieties
Let $\Delta$ be a polytope and consider the projective toric variety $P_{\Delta}.$ Given a curve $C \subset \mathbb{P}_{\Delta},$ which is not toric, is it true that in the Chow group we have $$ C = \...
- 65
2 votes
0 answers
178 views
Non-proper intersection between divisors on $\mathbb{P}^1$-bundle of Hirzebruch surfaces
We are working on algebraic closed field $k$. Let $\mathbb{F}_1$ be the Hirzebruch surface $\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(-1))$, $C_0$ and $C_{\infty}$ are its zero and infinity sections ...
- 41
2 votes
1 answer
164 views
Examples of nontrivial configurations of rational curves of degree $\leq 3$ in the projective plane
Consider the complex projective plane $P^2$. A rational curve in $P^2$ of degree $\leq 3$ is either a line, a smooth conic, a nodal cubic, or a cuspidal cubic. I am looking for some "nontrivial&...
- 427
0 votes
0 answers
171 views
How to prove that a specific quadric intersection is complete and irreducible?
Let's borrow the quadric intersection $I$ from another question. More precisely, let $k$ be an algebraically closed field of characteristic $\neq 2$ and $a_1, a_2, \cdots, a_n \in k^*$ be some ...
- 1,873
4 votes
0 answers
214 views
Is the group of homologically trivial cycles in a variety over a finite field torsion?
Let $X$ be a smooth projective variety over $\mathbb{F}_q$. Is any cycle in the Chow group $CH^i(X)$ which is trivial in $\ell$-adic cohomology automatically torsion? For abelian varieties I believe ...
- 571
2 votes
1 answer
274 views
Intersection in toric variety
In a toric variety $T$ of dimension $11$ I have a subvariety $W$ of which I would like to compute the dimension. On $T$ there is a nef but not ample divisor $D$ whose space of sections has dimension $...
- 281
1 vote
0 answers
137 views
Calculation of intersection multiplicity after the restricting to a fiber
Let $X\to\operatorname{Spec} \mathbb Z$ be an arithmetic surface which is projective, regular and integral. Let $D$ and $E$ two divisors intersecting at a point $x\in X$ that lies over the prime $p$. ...
- 159
4 votes
0 answers
163 views
Inverse direction of Hodge index theorem
The Hodge index theorem states that the intersection matrix associated to curves on a smooth algebraic surface has a specified signature---namely, if the intersection matrix has size $n \times n$ then ...
- 2,175
5 votes
1 answer
452 views
Contractibility of a curve on a surface
Let $X\subset \mathbb{P}^3$ be the surface defined by the equation $xy-zw=0$, and consider the curve $E \subset X$ defined by the equation $x=z=0$. Question. Can $E$ be contracted to a point?
- 368
3 votes
1 answer
583 views
Bloch–Beilinson conjecture for varieties over function fields of positive characteristic
Is there a version of the Bloch–Beilinson conjecture for smooth projective varieties over global fields of positive characteristic? The conjecture I’m referring to is the “recurring fantasy” on page 1 ...
- 571
4 votes
0 answers
256 views
Obstruction to finding a Whitney disk
Let $p,q\geq 3$ be integers. Let $M$ be a compact oriented smooth $(p+q)$-manifold and $P$ and $Q$ compact submanifolds of dimensions $p$ and $q$ intersecting transversely. Assume that $M,P$ and $Q$ ...
2 votes
0 answers
192 views
Is there a name for a normal, projective variety where every effective divisor is ample?
Is there a name for a normal, projective variety such that every effective divisor is ample? Examples of such varieties are projective space, weighted projective spaces, and simple Abelian varieties ...
- 1,179
3 votes
0 answers
193 views
Simple Grothendieck-Riemann-Roch computation with relative Todd class
$\DeclareMathOperator\Tot{Tot}\DeclareMathOperator\ch{ch}\DeclareMathOperator\td{td}\DeclareMathOperator\ker{ker}\DeclareMathOperator\rk{rk}$I was wondering if the following is correct: Let $X=\Tot(L)$...
- 139
1 vote
0 answers
293 views
Todd class of blow-up
Let $i:X\hookrightarrow Y$ be an embedding of two non-singular projective varieties over $\mathbb{C}$. Consider the blow-up $f:Y' = Bl_XY \to Y$, and the corresponding embedding $j:E\hookrightarrow Y'$...
- 91
0 votes
0 answers
300 views
Excision in "3264 and all that" by Eisenbud-Harris
In Proposition 1.14, page 25 in the book "3264 and all that Intersection Theory in Algebraic Geometry" the authors define a right exact sequence: $$ Z(\mathbb{P}^1 \times X) \rightarrow Z(X) ...
- 137
1 vote
1 answer
717 views
Very ample + effective = ample?
Sorry if this question is not appropriate for this site, but I haven't got an answer on stackexchange. It's well known that there are divisors (on a normal projective variety over the complex numbers) ...
- 195
1 vote
1 answer
136 views
Number of intersection points of a system of quartic polynomials induced by the projected matrix commutator
Let $A, B \in \mathbb{R}^{d \times d}$ denote two symmetric positive definite matrices. I am interested in solutions $V_r \in \mathbb{R}^{d \times r}, 1 < r < d$ to the system of quartic ...
- 11
3 votes
2 answers
507 views
Negative intersection number between curve and effective divisor
Let X be a complex projective variety and E be an irreducible effective divisor on it. Then, I want to know whether the following set is finite: {C | C be an irreducible curve and C.E<0}. I know ...
- 31
2 votes
0 answers
130 views
Reference for numerically non-negative polynomials for nef vector bundles
Let $K$ be a field. A polynomial $F \in \mathbb{Q}[X_1, \dots, X_r]$ which is weighted homogeneous of degree $n$ with respect to the grading $\deg(X_k) = k$ is called numerically non-negative for nef ...
user513306
1 vote
1 answer
149 views
On the situation of intersections along a proper morphism
The short question is: Say $p:\bar{X}\rightarrow S$ is a proper and normal morphism with the following properties: S is integral and smooth over a certain base field $k$, $\bar{X}$ has a smooth and ...
- 13
1 vote
0 answers
164 views
Multiplicity and the perfect projective line
Let $\mathbf{F}_p$ be the field with $p$ elements, and $X = (\mathbf{P}^1_{\overline{\mathbf{F}}_p})^\text{perf}$ the inverse perfection of the projective line over $\mathbf{F}_p$. Let $\Gamma$ be the ...
- 85
1 vote
1 answer
547 views
Self-intersection of the diagonal on a surface
Let $X$ be a smooth projective curve over the complex numbers, and take $\Delta$ the diagonal divisor on $X\times X$. Using the adjunction formula, one computes $\Delta\cdot\Delta =2-2g$ for $g$ the ...
- 85
1 vote
0 answers
336 views
Pull and push formula for degree for non-flat morphism
Let $\varphi\colon X_1\to X_2$ be dominant proper morphism of finite degree (in particular $\dim X_1=\dim X_2$) between varieties. Let $D \subset X_2$ be a Cartier divisor. Is it true that $$\varphi_*...
- 229
3 votes
0 answers
478 views
Why is the "wrong" definition of intersection of varieties the "right" one for generalized Bézout?
For ease of notation, define the degree of a variety to be the sum of the degrees of its irreducible components. The generalized Bézout theorem (due to Fulton and Macpherson) states that, for $V_1$, $...
- 21.7k
2 votes
1 answer
166 views
Sufficient condition for pair of real quadrics to have real intersection
In the following, when I talk about the zero of a homogeneous polynomial I always mean a projective zero. Let $ q $ be a real quadric. Then $ q $ has a real zero if and only if $ q $ has indefinite ...
- 4,577