Questions tagged [intersection-theory]

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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....
Stefan Vujic's user avatar
2 votes
0 answers
129 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 ...
Schemer1's user avatar
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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 $X=\Tot(L)$...
Simonsays's user avatar
  • 139
1 vote
0 answers
168 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 $j:E\hookrightarrow Y'$...
locallito's user avatar
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0 answers
217 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) ...
Andarrkor's user avatar
  • 109
1 vote
1 answer
250 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) ...
Calculus101's user avatar
1 vote
1 answer
60 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 ...
mtcli's user avatar
  • 11
3 votes
2 answers
348 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 ...
Mountain's user avatar
2 votes
0 answers
104 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 ...
user avatar
1 vote
1 answer
111 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 ...
Lee Peilin's user avatar
1 vote
0 answers
141 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 ...
Tim's user avatar
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1 vote
1 answer
290 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 ...
Tim's user avatar
  • 85
1 vote
0 answers
196 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_*...
Galois group's user avatar
3 votes
0 answers
355 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 $V_1$, $...
H A Helfgott's user avatar
  • 19.3k
2 votes
1 answer
89 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 ...
Ian Gershon Teixeira's user avatar
5 votes
0 answers
131 views

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 $...
timaeus's user avatar
  • 171
1 vote
1 answer
120 views

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)= ...
red_trumpet's user avatar
  • 1,061
1 vote
0 answers
231 views

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 ...
WakeUp-X.Liu's user avatar
1 vote
0 answers
99 views

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 ...
Dave's user avatar
  • 111
3 votes
0 answers
165 views

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 (...
M-M's user avatar
  • 31
2 votes
1 answer
202 views

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 $X\times X$ ...
user avatar
2 votes
0 answers
73 views

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{...
blancket's user avatar
  • 161
2 votes
1 answer
442 views

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 ...
Utf's user avatar
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0 answers
107 views

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 $\mathbb{Q}$-...
Boris's user avatar
  • 501
1 vote
0 answers
70 views

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 $(H,0)\subset(\mathbb{C}^{n+1},0)$ ...
MathBug's user avatar
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3 votes
0 answers
164 views

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 ...
user avatar
1 vote
0 answers
93 views

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, ...
user1131059's user avatar
11 votes
2 answers
519 views

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 ...
Puzzled's user avatar
  • 8,832
2 votes
0 answers
81 views

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 ...
Adterram's user avatar
  • 1,361
1 vote
1 answer
286 views

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 ...
user267839's user avatar
  • 5,938
2 votes
1 answer
111 views

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 ...
RaffaeleScandone's user avatar
2 votes
0 answers
168 views

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 ...
S.D.'s user avatar
  • 492
6 votes
2 answers
253 views

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 $\text{Tor}_n^R(R/I, R/J)$ ...
Alex's user avatar
  • 313
4 votes
1 answer
193 views

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 ...
Doyoung Choi's user avatar
2 votes
1 answer
217 views

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 ...
Jackson Morrow's user avatar
9 votes
1 answer
384 views

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 ...
H A Helfgott's user avatar
  • 19.3k
1 vote
0 answers
65 views

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 ...
Doyoung Choi's user avatar
4 votes
1 answer
130 views

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 ...
Jérémy Blanc's user avatar
1 vote
0 answers
76 views

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 $Y = \mathbb{P}(\mathcal{L}^{\oplus 3})$...
Doyoung Choi's user avatar
3 votes
1 answer
192 views

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 ...
Puzzled's user avatar
  • 8,832
2 votes
1 answer
188 views

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.
6666's user avatar
  • 343
2 votes
1 answer
236 views

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 ...
Roxana's user avatar
  • 519
1 vote
0 answers
245 views

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 $\...
Kim's user avatar
  • 505
6 votes
1 answer
456 views

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 ...
curious math guy's user avatar
2 votes
0 answers
116 views

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-...
Ben's user avatar
  • 1,010
2 votes
0 answers
93 views

$0$-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(...
Puzzled's user avatar
  • 8,832
3 votes
2 answers
189 views

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 ...
user45397's user avatar
  • 2,195
3 votes
1 answer
190 views

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 $...
user45397's user avatar
  • 2,195
2 votes
0 answers
176 views

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 $2$. ...
Jana's user avatar
  • 2,022
4 votes
2 answers
306 views

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 ...
Ben's user avatar
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