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2
votes
1answer
20 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
35 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
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
votes
1answer
84 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
31 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
58 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
43 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
41 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
26 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
35 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
38 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
30 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
37 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
57 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
80 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
27 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
58 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
47 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
46 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
39 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
50 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
133 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
74 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
108 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
49 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
43 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
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
votes
1answer
187 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
44 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
42 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
64 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
36 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
57 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
votes
1answer
118 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
36 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
vote
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
66 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
77 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, ...
1
vote
0answers
41 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 ...
3
votes
1answer
110 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 $L_i$ ...
18
votes
0answers
460 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": $S = \{(x,y) : |x|^p + |y|^p = 1\}$, ...
1
vote
0answers
43 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
votes
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
votes
1answer
72 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 ...
6
votes
1answer
184 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 ...
4
votes
1answer
276 views

Linear equivalence of Divisors on a Surface

Studying algebraic geometry, while the abstract theory is pretty clear to me I often feel puzzled in practice. Here I am trying to understand linear equivalence of divisors in some practical ...
3
votes
0answers
101 views

When is an intersection of varieties finite

Consider the general Bezout's theorem: If $p_1 \ldots p_n$ are polynomials with degrees $d_1,\ldots, d_n$ in $\mathbb{R}[x_1,\ldots,x_n]$, with $V = \{a=(a_1,\ldots, a_n) | p_i(a) = 0, \forall i\}$ ...
3
votes
3answers
268 views

Transverse intersection of multiple submanifolds

Let $M$ be a smooth manifold and suppose that we have three (or more) submanifolds $N_1,N_2,N_3\subset M$. What is the right notion of "transverse intersection" of $N_1,N_2,N_3$, i.e. what is the ...
5
votes
0answers
89 views

Why is an intersection product $X.C=0$?

Here is the situation: $S$ is a nonsingular complex projective surface, and $C=X\cup_AY\subset S$ is a uninodal curve of compact type: it is obtained by glueing two nonsingular curves $X,Y\subset S$ ...
6
votes
1answer
69 views

Injectivity of the push-forward of cycles modulo rational equivalence

Let $Y$ be a closed subscheme of a scheme $X$, and let $i: Y \rightarrow X$ be the inclusion morphism. We have the push-forward $i_{\ast}: A_{k}(Y) \rightarrow A_{k}(X)$, where $A_{k}(X)$ denotes the ...