All Questions
Tagged with intersection-theory ag.algebraic-geometry
329 questions
3
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168
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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)...
- 658
3
votes
0
answers
148
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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 ...
- 658
4
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1
answer
286
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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 ...
- 149k
2
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1
answer
663
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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 ...
CommunityBot
- 1
1
vote
0
answers
80
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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 $...
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1
vote
1
answer
141
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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 ...
- 149k
2
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1
answer
265
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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 = \...
- 149k
2
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0
answers
149
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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 ...
- 63.9k
2
votes
1
answer
114
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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&...
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0
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0
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110
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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 ...
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4
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0
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167
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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 ...
- 531
2
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1
answer
183
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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 $...
- 149k
1
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0
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104
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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$. ...
- 12.9k
3
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1
answer
370
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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 ...
- 1,404
4
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0
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137
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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
3
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1
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255
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About the contractability
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?
- 66.3k
2
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0
answers
145
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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 ...
- 5,429
3
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0
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148
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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)$...
- 63.9k
1
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0
answers
207
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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
2
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1
answer
116
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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 ...
- 341
0
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0
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239
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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) ...
- 109
1
vote
1
answer
500
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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) ...
- 6,262
1
vote
1
answer
539
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Hilbert polynomial as function of the Segre classes
Let $X\subset\mathbb{P} ^ N$ be a smooth irreducible complex projective variety of dimension 3 (or better yet, dimension $n$).
Is it possible to express the Hilbert polynomial of $X$ as a function of ...
- 1,446
3
votes
2
answers
436
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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 ...
- 28.8k
2
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0
answers
109
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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 ...
- 63.9k
1
vote
1
answer
121
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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 ...
- 149k
1
vote
0
answers
145
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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 ...
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1
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1
answer
370
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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 ...
- 28.8k
1
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0
answers
243
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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_*...
- 219
3
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0
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393
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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 $V_1$, $...
- 20.2k
1
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1
answer
141
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Are horizontal divisors on abelian fibered hyperkähler manifolds proportional in $NS(X)$ up to vertical divisors?
Oguiso writes[1]
Theorem 1.1 Let $f: X \to \mathbf P^n$ be an abelian fibered HK [hyperkähler] manifold. Let $K = \mathbf C(\mathbf P^n)$ and let $A_k$ be the generic fiber of $f$. Then, $\rho(A_K)= ...
- 38k
17
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2
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2k
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What does taking the graded algebra do to the Grothendieck group, and its relation to the Chow ring?
Let $X$ be a nonsingular variety. (Perhaps some/all of this works over more general smooth schemes, but let's stick to the simple case.)
In, e.g., Fulton's Intersection Theory chapter 15, and Soule's ...
- 156k
1
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0
answers
290
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Is there some relations between derived category and intersection theory?
After learning the traditional intersection theory (W. Fulton's Intersection Theory and D. Eisenbud & J. Harris's 3264 and all that), I have some biased thinking about what I have learned in this ...
- 63.9k
1
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0
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103
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Is $K_F\cdot C\leq K_X\cdot C$ for a fibre $F\subseteq X$ containing the curve $C$?
This is a question that I originally posted on Math Stack Exchange. After a couple of days I have not received any comments or answers, and after thinking about it more I realize that this question is ...
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2
votes
1
answer
243
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Finite flat pullback of the diagonal
Let $X, Y$ be smooth projective connected complex varieties of the same pure dimension $d$ and $f : X\to Y$ a finite flat surjective morphism.
Let $\Delta_X$ be the closed subscheme of $X\times X$ ...
- 28.8k
2
votes
1
answer
642
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How to show that the intersection of two certain affine varieties is reduced?
$\DeclareMathOperator\codim{codim}$Let $X=V(I)$, $Y=V(J)$ be two affine varieties. I'd like to know possibile strategies to understand when their schematic intersection, i.e. $X\cap Y=V(I+J)$, is ...
- 12.9k
0
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0
answers
137
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Intersection product of $\mathbb{Q}$-Cartier divisors with irreducible complete curves is well-defined
I am learning the notion of intersection product of a $\mathbb{Q}$-Cartier divisor with an irreducible complete curve on a normal variety. The definition I learned is that if $D$ is a $\mathbb{Q}$-...
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1
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0
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70
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Prescribed intersection of varieties
Every variety here is complex analytic, or complex algebraic if it solves anything.
Given a germ of a (possibly singular, nor necessarily irreducible) hypersurface $(H,0)\subset(\mathbb{C}^{n+1},0)$ ...
- 333
3
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0
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174
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Intersection theory on schemes with Gorenstein singularities
Is there a good reference/book on Intersection theory on schemes with Gorenstein singularities? Does the construction of the intersection of cycles discussed in Fulton's book also hold for schemes ...
CommunityBot
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11
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2
answers
558
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Hypersurface of singular plane cubics
In the projective space $\mathbb{P}^9 = \mathbb{P}(\mathbb{C}[x,y,z]_3)$, parametrizing plane cubics, consider the hypersurface $X\subset\mathbb{P}^9$ whose points corresponds to singular cubics. The ...
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11
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4
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4k
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Question on Kähler/ample cone, cone of curves....
Assume $X$ is smooth "simply connected" complex projective variety and $Y\subset X$ a smooth hyperplane section. ( $Y= X\cap H$, $H\subset \mathbb{P}^n$).
Let's $NE(X)$ be the cone of effective 1-...
- 35.5k
1
vote
1
answer
431
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Blow-up of a pencil of cubic curves (from Miranda's basic theory of elliptic surfaces)
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 that I do not understand yet.
Let $C_1$ be a smooth ...
- 66.3k
15
votes
3
answers
2k
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A nontrivial surface on which any two curves intersect
One interesting property of the projective plane is that any two plane curves intersect. (More generally, if $V$ and $W$ are subvarieties of any projective space, and codim $V$ + codim $W \geq 0$, ...
- 12.4k
2
votes
1
answer
243
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Intersection of translate of divisors on abelian variety
Setup. Let $K$ be an algebraically closed field of characteristic zero, and let $A/K$ be a simple abelian variety of dimension $n$. Let $\{ x_1,x_2,\dots,x_{m^{2n}}\}$ denote the $m$-torsion points of ...
- 254
4
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2
answers
328
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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 ...
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2
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0
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172
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Intersection theory on normal crossing algebraic surfaces
Let $X$ be an algebraic surface with normal crossing singularities. Suppose the singular locus of $X$ is a smooth curve. Let us denote it by $C$. Suppose $D$ is another smooth curve in $X$ which ...
- 494
4
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1
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220
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About Fulton's Intersection theory Appendix Lemma A 4.1
The assumption for Lemma A.4.1 is $A \to B$ is flat. The second assumption is that $A$ and $B$ are Artinian rings. From this Lemma A.4.1 states that $l_B(B) = l_A(A) \cdot l_B(B/mB)$ where $m$ is the ...
- 12.9k
10
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1
answer
424
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To whom is Bézout's theorem for varieties due?
The following is a modern, fairly general form of Bézout's theorem. (There are forms that are more general and/or more precise; bear with me.)
Define the degree of a reducible variety to be the sum of ...
- 455
1
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0
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71
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Comparison between residual intersection in Fulton's intersection theory and Aluffi's result on Milnor class
$\textbf{Question}$ I deduced that $m(A \cup B, V) = 0 $ for nonsingular variety $V$ and nonsingular hyper surfaces $A$ and $B$ whose intersection is also nonsingular. But I do not think it is true ...
- 121
4
votes
1
answer
134
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Isomorphism outside of negative curves against the canonical
Let $X$ be a smooth projective complex variety and let us suppose that the closure of the union of curves $C$ on $X$ that are non-positive against the canonical divisor is a closed subset $F\subsetneq ...
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