Skip to main content
MathOverflow

Return to Question

Minor changes to formatting.
Source Link
Sean Lawton
Sean Lawton
  • 8.5k
  • 3
  • 46
  • 78

In this question a variety means any subset of complex projective space $\mathbb CP^n$ that is the set of common zeroes of a set of homogeneous polynomials. Thus if $A,B\subset{\mathbb C}P^n$ are varieties so is $A\cup B$. 

Suppose $V_k$ is a sequence of varieties that converge to a subset $V\subset{\mathbb C}P^n$ in the Hausdorff topology on closed subsets of ${\mathbb C}P^n$. (This means after choosing a Riemannian metric on ${\mathbb C}P^n$ that given $\epsilon>0$ there is $K$ so that if $k>K$ then $V\subset N_{\epsilon}(V_k)$ and $V_k\subset N_{\epsilon}(V)$ where $N_{\epsilon}(X)$ is an $\epsilon$-neighborhood of $X$) Also.

Also suppose that there is $d$ such that each $V_k$ is the set of common zeroes of some polynomials of degree at most $d$.   

Is it true that $V$ is always a variety ?. 

This is a theorem of Ed Bishop if the $V_k$ are pure-dimensional.

In this question a variety means any subset of complex projective space $\mathbb CP^n$ that is the set of common zeroes of a set of homogeneous polynomials. Thus if $A,B\subset{\mathbb C}P^n$ are varieties so is $A\cup B$. Suppose $V_k$ is a sequence of varieties that converge to a subset $V\subset{\mathbb C}P^n$ in the Hausdorff topology on closed subsets of ${\mathbb C}P^n$. (This means after choosing a Riemannian metric on ${\mathbb C}P^n$ that given $\epsilon>0$ there is $K$ so that if $k>K$ then $V\subset N_{\epsilon}(V_k)$ and $V_k\subset N_{\epsilon}(V)$ where $N_{\epsilon}(X)$ is an $\epsilon$-neighborhood of $X$) Also suppose that there is $d$ such that each $V_k$ is the set of common zeroes of some polynomials of degree at most $d$.  Is it true that $V$ is always a variety ?. This is a theorem of Ed Bishop if the $V_k$ are pure-dimensional.

In this question a variety means any subset of complex projective space $\mathbb CP^n$ that is the set of common zeroes of a set of homogeneous polynomials. Thus if $A,B\subset{\mathbb C}P^n$ are varieties so is $A\cup B$. 

Suppose $V_k$ is a sequence of varieties that converge to a subset $V\subset{\mathbb C}P^n$ in the Hausdorff topology on closed subsets of ${\mathbb C}P^n$. (This means after choosing a Riemannian metric on ${\mathbb C}P^n$ that given $\epsilon>0$ there is $K$ so that if $k>K$ then $V\subset N_{\epsilon}(V_k)$ and $V_k\subset N_{\epsilon}(V)$ where $N_{\epsilon}(X)$ is an $\epsilon$-neighborhood of $X$).

Also suppose that there is $d$ such that each $V_k$ is the set of common zeroes of some polynomials of degree at most $d$. 

Is it true that $V$ is always a variety ? 

This is a theorem of Ed Bishop if the $V_k$ are pure-dimensional.

defined Hausdorff topology
Source Link
Daryl Cooper
Daryl Cooper
  • 700
  • 4
  • 11

In this question a variety means any subset of complex projective space $\mathbb CP^n$ that is the set of common zeroes of a set of homogeneous polynomials. Thus if $A,B\subset{\mathbb C}P^n$ are varieties so is $A\cup B$. Suppose $V_k$ is a sequence of varieties that converge to a subset $V\subset{\mathbb C}P^n$ in the Hausdorff topologyHausdorff topology on closed subsets of ${\mathbb C}P^n$. Also(This means after choosing a Riemannian metric on ${\mathbb C}P^n$ that given $\epsilon>0$ there is $K$ so that if $k>K$ then $V\subset N_{\epsilon}(V_k)$ and $V_k\subset N_{\epsilon}(V)$ where $N_{\epsilon}(X)$ is an $\epsilon$-neighborhood of $X$) Also suppose that there is $d$ such that each $V_k$ is the set of common zeroes of some polynomials of degree at most $d$. Is it true that $V$ is always a variety ?. This is a theorem of Ed Bishop if the $V_k$ are pure-dimensional.

In this question a variety means any subset of complex projective space $\mathbb CP^n$ that is the set of common zeroes of a set of homogeneous polynomials. Thus if $A,B\subset{\mathbb C}P^n$ are varieties so is $A\cup B$. Suppose $V_k$ is a sequence of varieties that converge to a subset $V\subset{\mathbb C}P^n$ in the Hausdorff topology on closed subsets of ${\mathbb C}P^n$. Also suppose that there is $d$ such that each $V_k$ is the set of common zeroes of some polynomials of degree at most $d$. Is it true that $V$ is always a variety ?. This is a theorem of Ed Bishop if the $V_k$ are pure-dimensional.

In this question a variety means any subset of complex projective space $\mathbb CP^n$ that is the set of common zeroes of a set of homogeneous polynomials. Thus if $A,B\subset{\mathbb C}P^n$ are varieties so is $A\cup B$. Suppose $V_k$ is a sequence of varieties that converge to a subset $V\subset{\mathbb C}P^n$ in the Hausdorff topology on closed subsets of ${\mathbb C}P^n$. (This means after choosing a Riemannian metric on ${\mathbb C}P^n$ that given $\epsilon>0$ there is $K$ so that if $k>K$ then $V\subset N_{\epsilon}(V_k)$ and $V_k\subset N_{\epsilon}(V)$ where $N_{\epsilon}(X)$ is an $\epsilon$-neighborhood of $X$) Also suppose that there is $d$ such that each $V_k$ is the set of common zeroes of some polynomials of degree at most $d$. Is it true that $V$ is always a variety ?. This is a theorem of Ed Bishop if the $V_k$ are pure-dimensional.

Source Link
Daryl Cooper
Daryl Cooper
  • 700
  • 4
  • 11

Limits of complex projective varieties in the Hausdorff topology on closed subsets of CP^n

In this question a variety means any subset of complex projective space $\mathbb CP^n$ that is the set of common zeroes of a set of homogeneous polynomials. Thus if $A,B\subset{\mathbb C}P^n$ are varieties so is $A\cup B$. Suppose $V_k$ is a sequence of varieties that converge to a subset $V\subset{\mathbb C}P^n$ in the Hausdorff topology on closed subsets of ${\mathbb C}P^n$. Also suppose that there is $d$ such that each $V_k$ is the set of common zeroes of some polynomials of degree at most $d$. Is it true that $V$ is always a variety ?. This is a theorem of Ed Bishop if the $V_k$ are pure-dimensional.