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37 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
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1answer
41 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
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1answer
26 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
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2answers
68 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
8 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
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1answer
22 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
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0answers
45 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: ...
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0answers
51 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: ...
5
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1answer
101 views

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

Is it necessary to read SGA VI to understand "Intersection Theory" by William Fulton?
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0answers
33 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
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1answer
61 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
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0answers
48 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^*} ...
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1answer
43 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 ...
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0answers
27 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
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0answers
39 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
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0answers
46 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
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0answers
35 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
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0answers
55 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
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1answer
63 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
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1answer
101 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
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0answers
29 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
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1answer
63 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 ...
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0answers
48 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 ...
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0answers
47 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: ...
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1answer
46 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
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1answer
60 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 $g(D)\geq3$ ?
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0answers
154 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
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1answer
82 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
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1answer
128 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
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0answers
52 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
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1answer
52 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 ...
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0answers
47 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
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1answer
216 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
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0answers
46 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 ...
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0answers
48 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
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0answers
73 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
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0answers
37 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
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1answer
62 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 ...
7
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1answer
132 views

Projection formula in Fulton's book

I have another question about intersection theory "a la Fulton". This is about the projection formula, that states that if you have $$\require{AMScd} \begin{CD} {X'} @>{\pi'}>> {Y'}\\ ...
2
votes
0answers
37 views

What is $\int_\gamma c_i(E)$ counting?

Let $M$ denote the Kontsevich moduli space of stable maps $\overline M_{0,n}(X,\beta)$, where $\beta\in A_1(X)$ and $X$ is a convex variety. I am trying to understand why $$\dim\, ...
1
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0answers
47 views

$C \times y_1, C \times y_2$ are numerically equivalent in $C \times Y$ for curve $C$.

This question comes from my former question. Let me formulate the question more precisely. Let $C$ be a projective smooth curve over an algebraically closed field. Let $Y$ be any variety. Why for ...
4
votes
0answers
68 views

Negative self intersection and section of the conormal sheaf for a singular complex curve

Let $M$ be a complex $2$-dimensional manifold and $C$ be a compact complex curve in $M$ (possibly singular). Let us suppose that there exists a holomorphic function $f\in\mathcal{O}(M)$ such that ...
0
votes
1answer
79 views

Continuous Deformation of Hypersurfaces

I'm reading Katz "Enumerative Geometry and String Theory". He shows that a degree $d$ hypersurface has cohomology class $dH $(where $H$ is the class of a hyperplane) by the following argument (p.80, ...
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0answers
42 views

Intersection between cycles and dominant maps

i must say in advance that i'm not very familiar with algebraic cycles and intersection theory, so i hope my question is not too trivial. Let $X$ and $Y$ be two K3 surfaces. Let ...
4
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1answer
115 views

Intersections of tangents with cubic are colinear

I am trying to do Exercise 5.33 on page 64 of Fulton's book on algebraic curves. 5.33 Let $C$ be an irreducible cubic, $L$ a line such that $L\bullet C = P_1+ P_2 + P_3,$ $P_i$ distinct. Let $L_i$ ...
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0answers
488 views

How many points of intersection between an ellipse and an $L_p$-circle?

Consider an ellipse $E$ in the plane, centered at the origin. (In my case, the minor axis points into the nonnegative quadrant.) Let S be an "$L_p$-circle": $S = \{(x,y) : |x|^p + |y|^p = 1\}$, ...
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0answers
44 views

Hypersurface containing a component of a variety of higher degree.

Let $X \subset \mathbb{CP}^N$ a $k$-dimensional projective variety cut out by polynomials of degree $\leq d$ and $f_0,\cdots,f_N$ be homogeneous polynomials of degree $d$ without commom factors with ...
4
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2answers
82 views

Surfaces: behavior of Pluricanonical maps

Let $S$ be an algebraic smooth surface (over $\Bbb{C}$). Suppose we have some fibration $p\colon S\rightarrow C$ onto a smooth curve. Let $f$ be a fiber, a curve, say. Let $K$ be a canonical divisor ...
7
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1answer
77 views

Elliptic Surfaces: a naive question

Ground field $\Bbb{C}$. Algebraic category. Smooth surfaces. Let $S$ be a minimal elliptic surface $p:S\rightarrow C$ the elliptic fibration (general fiber = elliptic curve). Suppose the ...
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1answer
187 views

Dimension of the image of the morphism associated to a Divisor

Let $S$ be an algebraic smooth surface over $\Bbb{C}$. Let $D\in\mathrm{Div}(S)$ be such that the complete linear system $|D|$ is base-point free and suppose $h^0(D)=N+1$ with $N>0$. To $D$ is ...