Questions tagged [intersection-theory]
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362
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Three annulus intersection problem [migrated]
Recently I faced a problem about intersection of three annuli. Imagine having three annuli same dimensions and you put them next to each other into triangular shape like putting together three circles....
2
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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 ...
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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 $X=\Tot(L)$...
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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 $j:E\hookrightarrow Y'$...
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217
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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) ...
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250
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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) ...
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1
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60
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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 ...
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2
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348
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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 ...
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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 ...
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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 ...
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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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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 ...
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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_*...
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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 $V_1$, $...
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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 ...
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Geometric interpretation of pairing between bordism and cobordism
In page 448 of these notes, a pairing between bordism and cobordism
$$\langle \ ,\ \rangle: U^m(X)\otimes U_n(X)\rightarrow \Omega^U_{n-m}$$
is defined as follows. Assume $x\in U^m$ is represented by $...
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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)= ...
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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 ...
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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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Intersection of two modules (and sub-modules) under tensors
I have a (hopefully quick) question regarding an intersection of two tensor modules. Let $K$ be a field and $A, B$ finitely-generated modules over the Laurent series $K((X))$. Let $\tilde{A}$ be a (...
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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 $X\times X$ ...
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Arf invariant for characteristic surfaces in closed 4-manifolds depends on homeomorphism type
Let $X$ be a closed smooth oriented 4-manifold with $H_1(X;\Bbb Z)=0$. Then for a smoothly embedded orientable surface $F\subset X$ which is characteristic, there is a well-defined invariant $\text{...
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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 ...
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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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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 $(H,0)\subset(\mathbb{C}^{n+1},0)$ ...
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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 ...
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93
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Intersection of schubert varieties
Let $L_1$ and $L_2$ $\in$ $\mathbb{P}^4$ be two planes that intersect in exactly one point $Q$. Let $P_1 \in L_1$, $P_2 \in L_2$ points, such that $P_1 \neq Q \neq P_2$. Using the duality theorem, ...
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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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Positivity of the intersection form [closed]
Let $M^4$ be a closed orientable smooth manifold and
$$
I: H_2(M,\mathbb Z) \times H_2(M,\mathbb Z) \to \mathbb Z
$$
its intersection form. If $b^+ \ge 1$, where $b^+$ denotes the number of positive ...
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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 ...
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111
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Hausdorff dimension and non-empty intersections with lines
Let $A\subseteq [0,1]^d$, $d\geq 2$, a set with Hausdorff dimension $\operatorname{dim}_{\mathcal{H}}A=s$. What is the minimum $s$ (if any) which guarantee that $A$ has non-empty intersections with a ...
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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 ...
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If Serre's intersection multiplicity $\chi(R/I, R/J)$ equals $\operatorname{length}_R (R/(I+J))$, then are $R/I, R/J$ Cohen-Macaulay?
Let $(R,\mathfrak m)$ be a regular local ring. Let $I,J$ be proper ideals of $R$ such that $R/(I+J)$ has finite length i.e. $\sqrt{I+J}=\mathfrak m.$ Since $I+J$ annihilates $\text{Tor}_n^R(R/I, R/J)$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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How to calculate the divisor given by closure of subscheme
Let $X \subset \mathbb{P}^N$ be a nonsingular projective variety over algebraically closed field which is embedded by very ample line bundle $\mathcal{L}$. Let $Y = \mathbb{P}(\mathcal{L}^{\oplus 3})$...
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Varieties connected by curves in projective spaces of small dimension
Let $X\subset\mathbb{P}^N$ be an irreducible complex variety. Fix an integer $a\geq 2$ and call $P_a$ the following property: given $x_1,\dots,x_a\in X$ general points there exists an irreducible ...
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Intersecton form of complete smooth Toric surface
Given a complete smooth Toric surface (over $\mathbb C$), is its intersection form well-known? Or is there an algorithm to calculate it? Thanks in advance.
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Algebraic and homological equivalence relations for $0$-cycles
Let $X$ be a connected smooth projective variety. Let $Z_0(X)_{alg}$ be the group of $0$-cycles algebraically equivalent to $0$ and $Z_0(X)_{\hom}$ be the group of $0$-cycles homologically equivalent ...
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A question on the Chow group on stacks
Let $X$ be a separated Deligne-Mumford stack finite type over the ground field. Then there is a Chow group $A_*(X)$ of $X$ which is well-behaved under flat pull-back, defined as follows.
Let $\...
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Mordell conjecture over function fields
So I've read (for instance in the introduction to R.S de Jong's thesis ) that the naive adaptation of the proof of the Mordell conjecture over function fields fails, even using Arakelov intersection ...
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Intersection of plane with Segre
$\newcommand{\complex}{\mathbb{C}}$ Let $Seg \subseteq \complex^M \otimes \complex^N$ be the set of elements of the form $v \otimes w$. It is well-known that a general linear subspace of dimension $(M-...
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$0$-dimensional intersection in weighted projective space
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(...
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189
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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 ...
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190
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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
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177
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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 $2$. ...
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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 ...