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### intersection theory on proper algebraic spaces

I have a question about the second example in Hartshorne's Algebraic Geometry, Appendix B, section 3 (given by Hironaka?). It is an example of a compact complex Moishezon 3-fold $X$ which is not an ...

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### Can a curve intersect a given curve only at given points?

Clearly the question in the title has a positive answer for analytic (or smooth, or continuous ...) curves, but what about the algebraic category? More specifically, given an irreducible polynomial ...

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### Reference for the Hodge Bundle

For the purposes of this question, let the Hodge bundle $\lambda$ be the bundle on a fibration of abelian varieties $X\to B$ with fiber over $b\in B$ the space of 1-forms on $X_b$, or the pullback to ...

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### A composition of a finite morphisms with the transpose correspondence: is it the multiplication by the degree?

Let $X,Y$ be smooth varieties over a field $k$ (which in my case is perfect of finite characteristic $p$; we may also assume that $X,Y$ are connected); $s:Y\to X$ is a finite morphism of degree $d$. ...

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**1**answer

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### Upper bound on the number of intersections of algebraic manifolds with affine planes

Given an algebraic map $f: B^d \to \mathbb{R}$, from the unit ball of dimension $d$ to the real, let $Y = f^{-1}(0)$. Then it is always possible to find a smaller ball $B_r \subset B^d$ not ...

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### Intersection of subvarieties versus ranks of Chow groups modulo numerical equivalences

A nice property of $\mathbb P^n$ is:
Property 1: Two subvarieties $U,V$ such that $\operatorname{dim} U +\operatorname{dim} V \geq n$ always intersect.
(for example, any 2 curves in $\mathbb ...

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### Riemann-Roch and Grothendieck duality: general case of Fulton's example 18.3.19

Fulton's "Intersection theory" book contains the following fact (example 18.3.19):
Let $X$ be a Cohen-Macaulay scheme over a field. Assume $X$ can be imbedded in a smooth scheme (so it has a ...

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### A nontrivial surface on which any two curves intersect

One interesting property of the projective plane is that any two plane curves intersect. (More generally, if $V$ and $W$ are subvarieties of any projective space, and codim $V$ + codim $W \geq 0$, ...

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### Graphs as cycles and intersection theory

I'm guessing that the answer to this question is well-known, but I'm struggling to find anything to help me.
Let $X,Y$ be compact manifolds of dimension $n,m$ respectively. Let $f:X \to Y$ be a ...

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### Context for intersection theory

This is a pretty basic question. Hartshorne defines "intersection multiplicity" for any two divisors on a surface. Fulton has an impressive framework of generalizing this in his book (my understanding ...

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### Chow Ring of Moduli Space of Abelian Varieties

Is there a good reference for the structure of the Chow ring of $\mathcal{A}_g$, the moduli space of complex principally polarized abelian varieties? More generally, references for the intersection ...

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### On finite endomorphisms of $\mathbf{P}^r$

This question is basically on applying the Grothendieck-Riemann-Roch theorem to finding a formula for the push-forward of a line bundle on $\mathbf{P}^r$ under a certain morphism. Since I have a lot ...

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**2**answers

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### Family of Enriques surfaces and GRR, Part 2

As I mentioned in my previous post, I am studying the article Moduli of Enriques surfaces and Grothendieck-Riemann-Roch.
The Grothendieck-Riemann-Roch theorem is applied there to show that, for any ...

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### Link: Serre's intersection formula <-> Bloch-Quillen Thm / When only intersecting divisors, is there 'shorter' approach of proof known?

In very short:
When proving the equivalence of intersection theory constructed through (Milnor) K-sheaves and their product vs. defining the product via Serre's local multiplicity formula + moving, I ...

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### What do higher Chow groups mean?

Let $z^i(X, m)$ be the free abelian group generated by all codimension $i$ subvarieties on $X \times \Delta^m$ which intersect all faces $X \times \Delta^j$ properly for all j < m. Then, for each ...

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### Curves with negative self intersection in the product of two curves

I wonder if the following is known: Are there two compact curves C1 and C2 of genus>1 defined over complex numbers, such that their product contains infinite number of irreducible curves of negative ...