The intersection-theory tag has no wiki summary.

**5**

votes

**1**answer

239 views

### Axiomatic intersection theory

Is there an axiomatic intersection theory?
What I expect is something like:
An intersection theory is a functor from the category of schemes(or other spaces) to the category of algebras, with ...

**3**

votes

**0**answers

219 views

### Bezout's theorem for non-proper intersections?

Is there a version of Bézout's theorem for non-proper intersections?
For my specific problem, the setup is as follows: Let $P_1,P_2,P_3,P_4\in\mathbb{C}[z_1,z_2,z_3,z_4]$, and suppose that (as a ...

**3**

votes

**0**answers

156 views

### Reference request: Samuel's multiplicity and degree

I am looking for references for the following simple facts.
Let $Y\subset \mathbb{P}^n$ be a variety (or pure-dimensional algebraic set). For $P\in Y$ denote by $e_p(Y)$ the (Samuel's) multiplicity ...

**5**

votes

**0**answers

133 views

### Divisibility of all entries in an intersection form

What are situations where one can conclude that all entries of an intersection form are divisible by a fixed integer?
More precisely: $F \subset S$ is a proper connected (usually reducible) ...

**0**

votes

**0**answers

134 views

### Relation between the index of (the sum of) lattices in euclidean space, and their orthogonal complement.

I am trying to figure out something concerning the index of lattices.
The question came about after reading the paper of W.Fulton and B.Sturmfels, ("Intersection theorey on toric varieties"). To ...

**1**

vote

**0**answers

113 views

### Non-proper intersection of projective schemes

Let $X, Y$ be projective varieties in $\mathbb{P}^n$ for $n>10$. Assume that dimensions of $X,Y$ are greater than $n/2$. My first question is as follows:
Is there any criterion ...

**1**

vote

**1**answer

786 views

### euler class of the normal bundle and self intersection number [duplicate]

Let $S$ be a compact submanifold of $X$ smooth manifold. I know that $T_X|_S=T_S\oplus N_{S/X}$ where $N_{S/X}$ is the normal bundle. I have read that the euler class $e(N_{S/X})$ corresponds (via ...

**4**

votes

**1**answer

262 views

### Local model of virtual fundamental cycle

The following baby version of virtual fundamental cycle is well known:
Let $M\subset V$ be the zero locus of a section $s$ of a vector bandle $E \to V$, in general $s$ is not transversal to the zero ...

**6**

votes

**0**answers

233 views

### Flat morphisms whose fibers are affine spaces

Let $f:X \to Y$ be a flat morphism, such that each fiber is isomorphic to the affine space $\mathbb{A}^n$. Then is is true that $f$ is a Zariski affine bundle? If not, is it at least an ètale affine ...

**8**

votes

**1**answer

310 views

### Commutativity of the Chow ring in positive characteristic

I was looking in Ravi Vakil's notes on Intersection Theory, Class 20, where he introduces the bivariant intersection theory, in particular the Chow ring $A^\ast (X)$.
On p. 2, he writes the following ...

**2**

votes

**1**answer

256 views

### Lefschetz Fixpoint theorem for non-orientable manifolds

The classical lefschetz fixpoint theorem is stated for oriented compact manifolds $M$ and a smooth map $f:M\to M$ as follows:
the intersection number $I(\Delta, \mathrm{graph}(f))$ is equal to the ...

**5**

votes

**2**answers

430 views

### Top chern class under finite, unramified, dominant morphism

Situation: Let $\Bbbk$ be an algebraically closed field. Assume that $\pi:Y\to X$ is an finite, dominant, unramified morphism between nonsingular varieties of dimensions $n$. Let $d=\deg(\pi)$.
What ...

**8**

votes

**1**answer

347 views

### Schemes with no nonconstant maps to lower dimensional schemes

Fix an algebraically closed field $k$ (arbitrary characteristic), all schemes will be of finite type over $k$.
(Property *): I'm interested in (classes of) examples of schemes $X$ (irreducible, of ...

**3**

votes

**3**answers

595 views

### Cohomology of vector bundles via Intersection Theory

Let $X$ be a smooth projective variety over a fixed field $k$ (take $k = \mathbb{C}$ if necessary). For a vector bundle $E$ on $X$, $ch(E)$ will be in the Chow ring.
$\textbf{Question 1: }$ If ...

**3**

votes

**1**answer

384 views

### Genus of non-complete intersections

Suppose $X\subset \mathbb{P}_k^N$ a nonsingular curve is a complete intersection of hypersurfaces $F_1, \cdots, F_{N-1}$ (of degrees $d_1, \cdots, d_{N-1}$ resp). Then, we know that the canonical ...

**14**

votes

**0**answers

465 views

### Is there a functor of points approach to algebraic cycles and intersection theory?

Motivation
Most of the algebraic geometry I have done so far was concerned with group schemes (e.g., abelian schemes, tori, unipotent groups). In that part of the field the "functor of points POV" is ...

**6**

votes

**1**answer

1k views

### Self-intersection and the normal bundle

Let $X/k$ be a surface nonsingular and proper over an algebraically closed field $k$. Let $C \subset X$ be a nonsingular curve. Then it is clear that the self-intersection $(C \cdot C)_X$ is ...

**0**

votes

**0**answers

191 views

### non proper intersection

Let X and Y two smooth closed subschemes of a smooth projective scheme Z over a field.
Let $W:=X\cap Y$.
I suppose that W is non empty and that the intersection of X and Y is non proper, i.e
...

**2**

votes

**0**answers

191 views

### Segre class of cones and Base change of projective cones

I'm trying to work out a result in Fulton's intersection theory and I think I need the following basic result about base change of projective cones (whose support may not be the entire base scheme).
...

**3**

votes

**0**answers

137 views

### Topology of K3 as a sum of two abelian fibrations.

Let $E$ be a blow-up of $\mathbb{P}^2$ at 9-points in the bases locus of a pencil of elliptic curves (A $T^2$ fibration over $S^2$).
K3 surfaces is obtained by removing a fiber from two copies of $E$ ...

**7**

votes

**2**answers

742 views

### Examples of excess intersection theory?

Let $M$ be a smooth manifold of dimension $m$ and $\pi:E\rightarrow M$ a vector bundle of rank $e$. Given a section $s$ of the bundle $\pi:E\rightarrow M$, we expect that the zero locus $Z(s)$ of $s$ ...

**3**

votes

**1**answer

284 views

### Intersection form on quotient manifold

Let $E_{1},E_{2}$ be elliptic curves over $\mathbb{C}$. We denote by $\iota_{i}$ the translation by a 2-torsion point on $E_{i}$. Then $G=\mathbb{Z}/2\mathbb{Z}$ acts freely on the the product ...

**3**

votes

**2**answers

173 views

### is intersection of a curve and a family of curves generically constant as a scheme?

(everything below is defined over an algebraically closed field)
Let $D$ be a (smooth) surface, and let $X \subset T \times D$ be a flat family of curves on $D$, where $T$ is irreducible. Let $E$ be ...

**3**

votes

**1**answer

266 views

### examples of Chow rings of surfaces

Can somone provide me (articles/literature) with examples of Chow rings of surfaces?
(e.g. here: http://www.math.uni-bonn.de/people/huybrech/K3Global.pdf Chapter 9)
What I want is a list of (smooth ...

**0**

votes

**0**answers

212 views

### Intersections with divisors on moduli of curves

Let $\overline{\mathcal{M}}_{g,n}$ be the moduli stack of stable curves of genus $g$ with $n$ points.
Consider
$0 \neq \gamma \in Pic(\overline{\mathcal{M}}_{g,n})$
the first Chern class of a ...

**1**

vote

**1**answer

163 views

### Putting two complete varieties in a family over the projective line

Let $X$ and $Y$ be two proper varieties of dimension $n$ over a field $k$. I'm looking for "reasonable" conditions, under which, there exists a proper and dominant morphism $f:V\to \mathbb{P}^1_k$, ...

**2**

votes

**2**answers

534 views

### Non-vanishing of cup product in cohomology

Let $X$ be a smooth projective variety over $\mathbb{C}$, and let $\alpha \in H^{2k}(X)$ be an algebraic cohomology class. Let us fix an $l$ such that $H^l(X) \neq 0$ and $H^{l+2k}(X) \neq 0$.
The ...

**2**

votes

**0**answers

126 views

### What is known about the structure of $\mathbb Z[c_1(\mathcal O_V(1))]$ for a projective $\Bbbk$-variety $V$?

Motivation:
Following Fulton's Intersection Theory, the Chern class of an arbitrary algebraic $\Bbbk$-scheme $X$ can be constructed as follows. First, define the graded by codimension abelian group ...

**7**

votes

**3**answers

838 views

### Chern classes of a blow-up at a point

Let $X$ be a nonsingular projective variety over $\mathbb{C}$, and let $\widetilde{X}$ be the blow-up of X at a point $p\in X$.
What relationships exist between the degrees of the Chern classes of ...

**1**

vote

**1**answer

292 views

### Hilbert polynomial as function of the Segre classes

Let $X\subset\mathbb{P} ^ N$ be a smooth irreducible complex projective variety of dimension 3 (or better yet, dimension $n$).
Is it possible to express the Hilbert polynomial of $X$ as a function of ...

**4**

votes

**0**answers

244 views

### where to learn K-group of coherent sheaves modulo numerical equivalence?

I am trying to emerge from my complete ignorance about intersection theory.
I have a bias toward sheaves, so I like the idea of doing intersection theory with the K-group of coherent sheaves. From ...

**6**

votes

**1**answer

571 views

### Calculating chern numbers yields a contradiction, why?

I am really stuck on this one. Let $Y=\mathbb{P}^n$ be the complex projective space and let $\tilde Y$ be the blow-up of $Y$ along a linear subvariety $X$ of codimension $d$. We get the following ...

**1**

vote

**1**answer

261 views

### How many zero-constraints can be added to a subspace-restricted matrix before no solution exists?

I'm trying to develop an estimator for the concentration matrix of a Gaussian Graphical Model. I've become stuck in trying to find conditions for the estimator to exist. I have a sufficient ...

**2**

votes

**1**answer

209 views

### Intersection powers of the exceptional divisor (and the transform of a hyperplane)

In light of my previous question, I am interested in the following scenario: Let $\tilde Y$ be the blow-up of $Y=\mathbb{P}^n$ along a linear subvariety $X\subseteq Y$ of codimension $d$, i.e. ...

**4**

votes

**3**answers

718 views

### (Second) Chern class of projective space, blown up in a linear subvariety

I already asked the same question at stack exchange but got no response for quite a while, so I thought I'd ask here. I also know that this has a certain resemblance to this question, but I cannot ...

**2**

votes

**1**answer

219 views

### Are there polynomials (almost) all of whose intersection numbers are divisible by some integer?

I've been playing around with some basic intersection theory, and I've wondered the following:
For every two integers $n$ and $m$, and complex numbers $a_1,...,a_n$, are there polynomials ...

**3**

votes

**1**answer

290 views

### Are there n polynomials for which all intersection multiplicities are at least m?

I don't know whether this is known or not, but I was thinking of the following problem.
Let $n$ and $m$ be natural numbers. Are there $n$ polynomial $f_1,...,f_n\in \mathbb{C}[x]$, such that all of ...

**1**

vote

**2**answers

227 views

### Is the zero set of a equivariant polynomial map of minimal degree a union of linear subspaces?

Suppose that a finite group acts on two vector spaces $X$ and $Y$, and that $f:X\longrightarrow Y$ is an equivariant polynomial map which is homogeneous of degree $n$, and that there does not exist ...

**4**

votes

**2**answers

553 views

### Vanishing associated to a resolution of singularities

Let $\pi: V\to W$ be a resolution of singularities, let $E \subset V$ be the exceptional divisor, and let $F$ be a coherent sheaf such that $R^i\pi_*F=0$ for $i>0$.
Can we conclude that ...

**4**

votes

**1**answer

2k views

### How many points determine a line?

Consider the affine space $\mathbb C^n$ and then, because of reasons, compactify it to obtain the projective space $\mathbb P^n$. One of the most basic axioms or propositions of geometry is that ...

**5**

votes

**1**answer

637 views

### Example of cone of numerically effective curves which is not polyhedral

I think I have seen more than one reference in which the cone of numerically effective curves can be 'not polyhedral', i.e. with an infinite number of extremal rays
I cannot remember where I read ...

**1**

vote

**2**answers

250 views

### Numerically negative exceptional divisor on a surface.

Suppose $S$ is an algebraic surface (possibly projective) over an algebraically closed field $k$. Suppose $D_i$ are irreducible smooth curves (rational, if you want) with negative self-intersection ...

**3**

votes

**1**answer

174 views

### non degenerate quadratic form on the group of correspondences on an algebraic curve?

Hi,
Given two (smooth, projective) curves $X$ and $Y$ over a field $k$, define a correspondence to be a line
bundle $L$ on $X\times Y$. A trivial correspondence is a correspondence of the form ...

**0**

votes

**2**answers

354 views

### definition of group operation in elliptic curves

Hi,
Using the isomorphism between an elliptic curve $E$ and its $Pic_1(E)$ group, one can
easily give $E$ the structure of a group variety after choosing a point $O\in E$. The
operation that one gets ...

**12**

votes

**1**answer

780 views

### Bezout's Theorem for weighted homogeneous polynomials

Bezout's Theorem states that for two homogeneous polynomials $f(x,y,z), g(x,y,z)$ over an algebraically closed field of degrees $m,n$ respectively, such that the two polynomials do not share a common ...

**0**

votes

**1**answer

332 views

### Degree of a real algebraic variety and regular morphisms

I'm reading Fulton's "Intersection theory", which i need for some applied needs.
And i have two questions on general definition of degree used in Fulton.
1)Let us we have a real algebraic variety ...

**14**

votes

**1**answer

1k views

### Geometric examples of the Serre intersection formula

The Serre intersection formula, as an alternating sum of contributions from Tor-groups, is something that combines a lot of ingredients that I'm interested in, but I've never really felt that I have a ...

**0**

votes

**2**answers

597 views

### Dimension of affine variety

Assume that I have $k$ polynomials $f_1(x_1,\ldots x_n),f_2(x_1,\ldots x_n),\ldots f_k(x_1,\ldots x_n)$ in $n>k$ variables. Is it possible to calculate, ,i.e., does there exist a fast algorithm, ...

**1**

vote

**1**answer

180 views

### Intersection positivity for curves and surfaces

Let $X$ be a smooth complete variety over an algebraically closed field of dimension $\geq3$. Given a divisor $D_1$ on $X$ with $D_1 \cdot C>0$ for every curve $C \subset X$, and a divisor $D_2$ ...

**0**

votes

**1**answer

361 views

### Intersection of curves

Let $f(x,y)=0$ and $g(x,y)=0$ be curves in $\mathbb R^2$. Assume that the origin $(0,0)\in \mathbb R^2$ is a $d$-fold point of $f$ and an $e$-fold point of $g$, respectively. Let $f_d(x,y)$ be the sum ...