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2
votes
1answer
33 views

Self-intersection of a cycle

If $X$ is a smooth projective variety of dimension $2n$, and $V \subset X$ is a smooth subvariety of dimension $\geq n$, then $V \cdot V$ makes sense as a class as an element of the Chow ring ...
0
votes
1answer
42 views

Trouble doing polynomial interpolation

I need to do a polynomial interpolation of a set $N$ of experimental points; the functional form I have to use to interpolate is this: $$ f(x) = a + bx^2 + cx^4,$$ as you can see the coefficient that ...
2
votes
0answers
51 views

A question about intersection number on surfaces

This question is from the Qing Liu's book: Algebraic Geometry and Arithmetic Curves, Exercise 9.1.6. Let $X\to S$ be an arithmetic surface and $X_s$ a closed fiber. Let $C_1,...,C_m$ denote the ...
0
votes
0answers
35 views

Meaning of “local equation” of a divisor.

Let $X$ be a smooth surface and moreover let $C,D$ be two effective divisors of $X$. Hartshorne says (page 357) that $C$ and $D$ meet transversally if the local equations $f,g$ of $C,D$ at $P$ ...
3
votes
1answer
60 views

Calculating canonical divisor in product of projective spaces.

Let $X$ be an intersection of two divisors of bidegree $(a,b)$ and $(c,d)$ in $\mathbb{P^2}\times \mathbb{P^2}$. Then how can I find the canonical divisor $K_X$? I'm asking because I have no ...
0
votes
0answers
34 views

Basic question: degree of normal bundle is not self-intersection number

For $C$ a (possibly singular) curve on a nonsingular projective surface $X$, let's define $C^2=deg_C(\mathcal{O}_X(C))$. Why is it not the same as $deg_C(N_{X|C})$ when $C$ is singular? Why do ...
2
votes
1answer
47 views

Questions about intersection of linear varieties in a projective space

Let $X, Y, Z$ be linear varieties of dimension $r, s, t$ respectively in $\mathbb{P}^n$. If $r+s\ge n$, then $X\cap Y\neq \varnothing$. Furthermore, if $X\cap Y\neq \varnothing$, then $X\cap Y$ is a ...
3
votes
0answers
44 views

Blowing-up of a linear sub-space is Fano?

Consider $P=\mathbb{P}^3\times \mathbb{P}^1$ with coordinates $([x_0:x_1:x_2:x_3],[u:v])$ and let $\varepsilon:X\to P$ be the blow-up of $\mathbb{P}^2\cong A=(x_3=v=0)\subseteq P$, of pure codimension ...
0
votes
3answers
87 views

Possible Intersection of Intervals

Suppose there are two intervals, where one of them is fixed. Is there a way to calculate all possible intersections of the intervals as shown in the figure? ? Notice that because $a,b$ and $c$ are ...
0
votes
0answers
15 views

antiholomorphic involution action on resolved orbifold of tori

Consider the space $\tilde X= T^2 \times T^2 \times T^2 / G$ where $T^2$ is a torus and $G=Z_2 \times Z_2$ acts as $\theta_1:(z_1,z_2) \mapsto (-z_1,-z_2)$, $\theta_2:(z_2,z_3) \mapsto (-z_2,-z_3)$ ...
7
votes
2answers
68 views

rational quartic in $\mathbb{P}^3$

According to Hartshorne (exercise IV.6.1), a rational curve of degree 4 in $\mathbb{P}^3$ is contained in a unique smooth quadric surface. If this is the case, then it must define a divisor on it. My ...
0
votes
1answer
33 views

how to calculate these intersections without having to count all combinations

We have the following sets: $X= {(a,b,c,d) ∈S: b< c < d},$ $Y= {(a,b,c,d) ∈S: a< c < d},$ $Z= {(a,b,c,d) ∈S: a< b < d},$ $F= {(a,b,c,d) ∈S: a< b < c},$ Where each of ...
0
votes
0answers
56 views

Generalization of Bezout Theorem to many-hypersurface case in Hartshorne's setting

I try to follow the ideas in Hartshorne's Chapter 1, Section 7. Suppose we have algebraic sets $Y_1,...,Y_l$, I try to define their intersection number $I(Y_1,...,Y_l)$ to be the leading term of the ...
2
votes
1answer
42 views

Dimension of moduli of lines on quadric

What is the dimension of the moduli space of lines on a general quadric hypersurface in $\mathbb{P}^n$? Maybe the question is quite trivial, but different intuitive approaches (à la Italian algebraic ...
1
vote
1answer
41 views

Intersection product of submanifolds of complex manifolds - Selfintersection

I do not understand something about the intersection product. I'm kind of new to this topic, so please consider that. I write down everything we discussed in a lecture. We defined the intersection ...
3
votes
2answers
100 views

Definition of intersection multiplicity in Hartshorne VS Fulton for plane curves

In exercise I.5.2 in Harshorne there is the following definition of intersection multiplicity for two curves in $\mathbb{A}^2$: \begin{equation} \mathrm{length}_{\mathcal{O}_P}\mathcal{O}_P / (f,g) ...
3
votes
1answer
12 views

Containment of two varieties with a lot of intersection

Given a projective variety $X\subset \mathbb P^n$ and a curve $C\subset \mathbb P^n$, when can I conclude that $C\subset X$, from the fact that $C$ and $X$ have 'many' points in common. I.e., is there ...
2
votes
1answer
28 views

Number of intersections of two closed loops on a genus zero surface

I have stumbled onto the following fact and I am quite helpless in seeing why this is true (although I can agree intuitively). Let $M$ be a surface of genus zero (open or closed, with or without ...
0
votes
0answers
72 views

Probability of infinite intersections

While I was studying Probability and random processes I came across the following question. Say I have $A_1,A_2, \ldots, A_n$ events such that $A_i$ is in $E$ but not equal to $E$. What is: ...
0
votes
0answers
55 views

Why is this theorem important in fulton's book?

In Fulton algebraic curves book, we have the following proprieties which help us to find the intersection number of a pair of function at a given point. afterwards he states this theorem: ...
6
votes
1answer
149 views

What are the prerequisites for Fulton's “Intersection Theory”?

Is it necessary to read SGA VI to understand "Intersection Theory" by William Fulton?
1
vote
0answers
36 views

Calculating the intersection product in CH(X)

Let CH$(X)$ be the Chow-Ring of a projective,smooth variety with cycles modulo rational equivalence. Lets assume Kunneth-Formula holds. There is an intersection product CH$^a(X) \otimes $ CH$^b(X) ...
3
votes
1answer
72 views

Complete intersections are connected

I have been stuck on this exercise in Vakil's notes (and moved on hoping it would come to me later), and it seems to be useful for other results (for example, when expressing curves as complete ...
1
vote
0answers
50 views

Chow Groups of $\mathbb{P}^n$

I'm trying to see that $A_k(\mathbb{P}^n)=\mathbb{Z}$ for all $k$. I am trying to do this with induction on $n$, by applying the excision sequence $$A_k(Y) \xrightarrow{i_*} A_k(X) \xrightarrow{j^*} ...
1
vote
1answer
45 views

Loci of cubics and intersection theory

Could you help me to understand what does the calculation in the bottom of the image mean? From where does $\tau^*(\zeta)$ appear? It is page 48 from preprint 3264 & All That Intersection Theory ...
1
vote
0answers
30 views

Pushforwards of Chow rings

Briefly, which properties morphism should have to induce pushforward of Chow rings? It is said in both Hartshorne and 3264 and All That that it should be proper, but why not just closed? Let $Z(X)$ ...
2
votes
0answers
46 views

A pullback in terms of a pushforward of a intersection

Let $\pi: X \rightarrow Y$ be a morphism of varieties. Suppose $s: Y \rightarrow X$ is a section of $\pi$, that is, $\pi \circ s = id$. I came across with the following identity: $$s^{\ast}(s(Y)) = ...
1
vote
0answers
61 views

can one intersect a weil divisor and curve

On a projective (integral) variety $X$, is there a well defined notion of the intersection of a Weil divisor and a curve? I know that there is a definition of the intersection of a Cartier divisor and ...
0
votes
0answers
37 views

Intersection product and Euler form

Can anyone provide me with a proof or a reference for the following statement: For curves $C$ and $D$ on a smooth commutative projective surface $X$ over an algebraically closed field, the ...
0
votes
0answers
77 views

Proving that the Neron-Severi Group is of finite rank.

I need help proving that the Neron-Severi Group, $N^{1}(X)$ is a free abelian group of finite rank. I am reading Lazarsfeld book, "Positive in Algebraic Geometry I", and he is trying to prove it ...
5
votes
1answer
76 views

Transverse intersection in a compact manifold

Is it true that if $M$ is a compact manifold and $X,Y$ are submanifolds of $M$ which intersect transversely that the intersection $X\cap Y$ consists of finitely many points? I'm trying to understand ...
3
votes
1answer
112 views

Question on Intersection Theory of Effective Divisors

I am reading Section 1.1C of Lazarsfeld "Positivity in Algebraic Geometry I" and I need help understanding one line. On Page 17, Remark 1.1.13(iii), he says the following: If $D_1,..., D_n$ are ...
0
votes
0answers
32 views

intersection theory on projective space

Let $\mathcal{L} := O(1) \in Pic(\mathbb{P}^d)$ be considered as an element of $CH^1(\mathbb{P}^d)$. What is its $d$-fold power in $CH^d(\mathbb{P}^d) = \mathbb{Z}$?
3
votes
1answer
68 views

Pullback of the normal cone

The introduction to Fulton's Intersection Theory reads in part: To give an idea of the main thrust of the text, we sketch what we call the basic construction. [...] To a closed regular embedding ...
1
vote
0answers
50 views

Is there a relation for when a circle intersects more than half the perimeter/circumference of another circle?

Is there some nice formula or algoritm for determining when a circle "hides"/intersects more than half of the perimeter of another circle, in a circle-circle interaction. Example image: Two example ...
1
vote
0answers
52 views

Intersection Theory flat/proper commutation

I have a problem with proposition 1.7 from Fulton's book, "Intersection Theory", that states that if $f$ is a proper morphism between $k$-schemes, $X$ and $Y$, and if we make a flat base change $g: ...
1
vote
1answer
53 views

Fixed components of linear systems on K3 surfaces

On a K3 surface, let $D$ be an effective divisor with $D^2\geq0$. Let $$D\sim D'+\Delta$$ be its decomposition in moving part and fixed part, respectively. Let $\Gamma$ be a prime component of ...
2
votes
1answer
65 views

Intersections with elliptic curves on a K3 surface

This is a fairly simple question. Suppose $E$ is an elliptic curve on a K3 surface $X$. Can we say that $E$ must intersect any curve $D\subset X$ of genus $g(D)\geq3$ ?
2
votes
0answers
177 views

Grassmannian bundle: any good reference?

I have met the notion of Grassmannian bundle of a vector bundle over a variety in intersection theory, but anywhere I look I just find a brief recall of how the stalks look like (my references so far ...
1
vote
1answer
98 views

Linear systems and hyperplane sections on surfaces

Let $S$ be a smooth projective surface. If $H$ is an hyperplane section on $S$ and $D$ a divisor (that can be not effective) such that $(H.D)<0$, why can we conclude that the linear system $|D|$ is ...
4
votes
1answer
155 views

Hyperplane sections on projective surfaces

I am studying Beauville's book "Complex Algebraic Surfaces". At page 2 he defines the intersection form (.) on the Picard group of a surface. For $L, L^\prime \in Pic(S)$ ...
0
votes
0answers
55 views

Auto-intersection of a line on a smooth cubic surface

Can someone help me with the following idea? I think that i made a mistake: Let $X$ be a smooth surface of degree $d$ in $\mathbb{P}^3$ and $L$ denote the divisor class of a line on $X$. We have ...
1
vote
1answer
59 views

Generators of the Chow ring of a space with a cellular decomposition

I am reading a paper "Algebraic varieties with rational dissections" by Chow. Here is a link: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC528229/?page=3 . A variety $V$ is said to have a rational ...
1
vote
0answers
53 views

The localization exact sequence.

Let $X$ a scheme and $Y$ a close subscheme and $i:Y \to X$ the inclusion. Let $U:=(X-Y)$ and $j:U \to X$ the inclusion map. If I denote with $CH(-)$ the Chow group, the sequence $$ CH_r(Y) ...
8
votes
1answer
262 views

Are these definitions of intersection multiplicity equivalent?

I am pretty sure the answer is yes. I normally work over $\mathbb{C}$ so i will do so here as well, to prevent myself from making silly mistakes. In projective space, one has Serre's famous ...
1
vote
0answers
48 views

A textbook reference for an Intersection Theory Result

I am using the following results: Given $M$ a smooth closed orientable manifold and $A$ and $B$ two smooth closed orientable submanifolds. $A$ and $B$ intersect transversely and $\dim A+\dim B=\dim ...
1
vote
0answers
60 views

Does this proof that the resultant provides an upper bound for intersection multiplicity look correct?

Let $f,g \in \mathbb{C}[x,y]$ such that $f(0,0)=g(0,0)=0$ and the varieties $V(f)$ and $V(g)$ are both smooth at $(0,0)$ such that the tangent line of $V(f)$ and $V(g)$ is not the $y$ axis. Let ...
3
votes
0answers
85 views

Reference request: generalized Lefschetz-Hopf fixed point theorem

Let $X$ be a smooth projective variety over $\mathbb{F}_p$. A basic (and very important) theorem is that we have equality $$ \# X(\mathbb{F}_{p^n}) = \sum_{i=0}^{2\dim X} (1)^i ...
3
votes
0answers
38 views

14 excess contribution?

I want to know the number of smooth conics that tangent 4 general lines and pass through 1 general point. Indeed, the number is 2. But from Bezout's theorem we get $1\cdot2^4=16$ on $\mathbb{P}^5$. ...
3
votes
1answer
66 views

Can the method of resolvents be used to give a proof of Bezout's Theorem?

Can the method of resolvents be used to give a proof of Bezout's Theorem? It seems to me like it should but I am unable to finish the proof. Here is what I have so far. Take two homogeneous ...