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10
votes
0answers
196 views

Smooth, complete varieties on which “zero is effective”

I will say zero is effective on a complete, smooth variety $X$ if some positive linear combination of irreducible varieties is rationally equivalent to zero. In other words, zero is effective if there ...
6
votes
0answers
139 views

Chow ring of extended tropicalizations

In Allermann-Rau '09, the authors define the Chow groups of an arbitrary abstract tropical cycle. In particular, one may take the tropical cycle to be the tropicalization of a subvariety of a torus. ...
1
vote
1answer
226 views

Nef classes on abelian varieties in positive characteristic

Thomas Bauer shows in http://arxiv.org/pdf/alg-geom/9712019v1.pdf that for a complex abelian variety a nef line bundle is numerically equivalent to an effective divisor (this is shown in Lemma 1.1). ...
3
votes
0answers
158 views

Chow ring of a $\mu_2$-gerbe

Suppose that $X$ is a stack, and $Y \to X$ is a $\mu_2$-gerbe. Is there any relationship between the integral Chow rings (in the sense of Edidin and Graham) of $X$ and $Y$? (I assume they become ...
6
votes
2answers
184 views

Action of an isomorphism in cohomology as the intersection with the class of the graph

Let $X$ and $Y$ be two complex manifolds of dimension 2 and let $\varphi:X\rightarrow Y$ be an isomorphism. I have read that the action of $\varphi^*:H^2(Y,\mathbb{Z})\rightarrow H^2(X,\mathbb{Z})$ ...
5
votes
1answer
244 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
0answers
222 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
0answers
158 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
0answers
135 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
0answers
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
0answers
116 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
1answer
857 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
1answer
276 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
0answers
242 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
1answer
318 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
1answer
267 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
2answers
434 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
1answer
351 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
3answers
618 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
1answer
397 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
0answers
481 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
1answer
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
0answers
198 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
0answers
192 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
0answers
138 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
2answers
770 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
1answer
286 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
2answers
174 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
1answer
295 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
0answers
214 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
1answer
165 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
2answers
541 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
0answers
133 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
3answers
874 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
1answer
301 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
0answers
248 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
1answer
576 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
1answer
262 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
1answer
211 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
3answers
743 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
1answer
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
1answer
291 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
2answers
231 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
2answers
555 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
1answer
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
1answer
660 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
2answers
255 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
1answer
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
2answers
358 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
1answer
797 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 ...