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edited Nov 27, 2012 at 17:07

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Here is a variation on Tom's argument (for the 1st question) that works: I will also work over ${\mathbb C}$, although, I do not think it is important. I will be proving that $n$ can be bounded depending only on dimension of ambient space and degree $d$ of the subvariety $V\subset P^N$. The proof is by induction on dimension $N$ of the ambient projective space. Everything is clear, if $N=1$ or if $dim(V)=1$. Suppose that we have a bound $n(d,N-1)$; consider a smooth subvariety $V\subset P^N$. Now, given any pair of points $p, q\in V$, I can find a projective hyperplane $H\subset P^N$ intersecting $V$ transversally (algebraic geometers, I think, would call it KleimanBertini's transversality, topologists would call it Sard's theorem). By Lefschetz, if $dim(V)\ge 2$, then $W=V\cap H$ is connected. For two generic projective hyperplanes $H_p, H_q$ through $p, q$, intersections $W_p=H_p\cap W, W_q=H_q\cap W$ are nonempty, smooth and connected. Now, we are done by induction applied to $W, W_p, W_q$ (connect $p$ to $x\in W\cap W_p$, then connect $x$ to $y\in W\cap W_q$, etc.).

Here is a variation on Tom's argument (for the 1st question) that works: I will also work over ${\mathbb C}$, although, I do not think it is important. I will be proving that $n$ can be bounded depending only on dimension of ambient space and degree $d$ of the subvariety $V\subset P^N$. The proof is by induction on dimension $N$ of the ambient projective space. Everything is clear, if $N=1$ or if $dim(V)=1$. Suppose that we have a bound $n(d,N-1)$; consider a smooth subvariety $V\subset P^N$. Now, given any pair of points $p, q\in V$, I can find a projective hyperplane $H\subset P^N$ intersecting $V$ transversally (algebraic geometers, I think, would call it Kleiman transversality, topologists would call it Sard's theorem). By Lefschetz, if $dim(V)\ge 2$, then $W=V\cap H$ is connected. For two generic projective hyperplanes $H_p, H_q$ through $p, q$, intersections $W_p=H_p\cap W, W_q=H_q\cap W$ are nonempty, smooth and connected. Now, we are done by induction applied to $W, W_p, W_q$ (connect $p$ to $x\in W\cap W_p$, then connect $x$ to $y\in W\cap W_q$, etc.).

Here is a variation on Tom's argument (for the 1st question) that works: I will also work over ${\mathbb C}$, although, I do not think it is important. I will be proving that $n$ can be bounded depending only on dimension of ambient space and degree $d$ of the subvariety $V\subset P^N$. The proof is by induction on dimension $N$ of the ambient projective space. Everything is clear, if $N=1$ or if $dim(V)=1$. Suppose that we have a bound $n(d,N-1)$; consider a smooth subvariety $V\subset P^N$. Now, given any pair of points $p, q\in V$, I can find a projective hyperplane $H\subset P^N$ intersecting $V$ transversally (algebraic geometers would call it Bertini's transversality, topologists would call it Sard's theorem). By Lefschetz, if $dim(V)\ge 2$, then $W=V\cap H$ is connected. For two generic projective hyperplanes $H_p, H_q$ through $p, q$, intersections $W_p=H_p\cap W, W_q=H_q\cap W$ are nonempty, smooth and connected. Now, we are done by induction applied to $W, W_p, W_q$ (connect $p$ to $x\in W\cap W_p$, then connect $x$ to $y\in W\cap W_q$, etc.).

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created Nov 27, 2012 at 16:55

Misha

31.2k
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94
163

Here is a variation on Tom's argument (for the 1st question) that works: I will also work over ${\mathbb C}$, although, I do not think it is important. I will be proving that $n$ can be bounded depending only on dimension of ambient space and degree $d$ of the subvariety $V\subset P^N$. The proof is by induction on dimension $N$ of the ambient projective space. Everything is clear, if $N=1$ or if $dim(V)=1$. Suppose that we have a bound $n(d,N-1)$; consider a smooth subvariety $V\subset P^N$. Now, given any pair of points $p, q\in V$, I can find a projective hyperplane $H\subset P^N$ intersecting $V$ transversally (algebraic geometers, I think, would call it Kleiman transversality, topologists would call it Sard's theorem). By Lefschetz, if $dim(V)\ge 2$, then $W=V\cap H$ is connected. For two generic projective hyperplanes $H_p, H_q$ through $p, q$, intersections $W_p=H_p\cap W, W_q=H_q\cap W$ are nonempty, smooth and connected. Now, we are done by induction applied to $W, W_p, W_q$ (connect $p$ to $x\in W\cap W_p$, then connect $x$ to $y\in W\cap W_q$, etc.).