Degree of Varieties and Segre's Embedding
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Let $X$ be an irreducible complex projective variety of dimension $p$ and consider two projective embeddings $iotacolon Xsubseteq mathbb{P}^n$ and $iota'colon Xsubseteq mathbb{P}^{n'}$. Denote by $Sigmacolon mathbb{P}^ntimesmathbb{P}^{n'}longrightarrow mathbb{P}^N$ the Segre embedding. Then we get another embedding $iota''colon Xsubseteqmathbb{P}^N$ given by $$xlongmapsto (iota(x),iota'(x))longmapstoSigmabig((iota(x),iota'(x)).$$ Assume that that $iota''colon Xsubseteqmathbb P^N$ has degree $d$. Then there exist $(p-1)$ linear functionals $theta''_1,ldots,theta''_{p-1}$ of $mathbb{C}^{N+1}$ such that the cardinality of $$lbrace theta''_1=cdots=theta''_{p-1}=0rbracecap iota''(X)$$ is equal to $d$. I would like to check that the functional $theta_i''$ above can be taken of the form $$theta_i''=theta_iotimestheta_i',$$ where $theta_i$ and $theta_i$' are linear functionals of $mathbb{C}^{n+1}$ and $mathbb{C}^{n'+1}$, respectively.
This a step (not proved) of Hoyt's proof of the topological invariance of the Chow monoid.
Any help is well accepted. Thanks in advance.
algebraic-geometry proof-writing
asked Dec 14 '18 at 8:16
Vincenzo ZaccaroVincenzo Zaccaro
1,254720
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add a comment |
$begingroup$
Let $X$ be an irreducible complex projective variety of dimension $p$ and consider two projective embeddings $iotacolon Xsubseteq mathbb{P}^n$ and $iota'colon Xsubseteq mathbb{P}^{n'}$. Denote by $Sigmacolon mathbb{P}^ntimesmathbb{P}^{n'}longrightarrow mathbb{P}^N$ the Segre embedding. Then we get another embedding $iota''colon Xsubseteqmathbb{P}^N$ given by $$xlongmapsto (iota(x),iota'(x))longmapstoSigmabig((iota(x),iota'(x)).$$ Assume that that $iota''colon Xsubseteqmathbb P^N$ has degree $d$. Then there exist $(p-1)$ linear functionals $theta''_1,ldots,theta''_{p-1}$ of $mathbb{C}^{N+1}$ such that the cardinality of $$lbrace theta''_1=cdots=theta''_{p-1}=0rbracecap iota''(X)$$ is equal to $d$. I would like to check that the functional $theta_i''$ above can be taken of the form $$theta_i''=theta_iotimestheta_i',$$ where $theta_i$ and $theta_i$' are linear functionals of $mathbb{C}^{n+1}$ and $mathbb{C}^{n'+1}$, respectively.
This a step (not proved) of Hoyt's proof of the topological invariance of the Chow monoid.
Any help is well accepted. Thanks in advance.
algebraic-geometry proof-writing
asked Dec 14 '18 at 8:16
Vincenzo ZaccaroVincenzo Zaccaro
1,254720
$endgroup$
add a comment |
$begingroup$
Let $X$ be an irreducible complex projective variety of dimension $p$ and consider two projective embeddings $iotacolon Xsubseteq mathbb{P}^n$ and $iota'colon Xsubseteq mathbb{P}^{n'}$. Denote by $Sigmacolon mathbb{P}^ntimesmathbb{P}^{n'}longrightarrow mathbb{P}^N$ the Segre embedding. Then we get another embedding $iota''colon Xsubseteqmathbb{P}^N$ given by $$xlongmapsto (iota(x),iota'(x))longmapstoSigmabig((iota(x),iota'(x)).$$ Assume that that $iota''colon Xsubseteqmathbb P^N$ has degree $d$. Then there exist $(p-1)$ linear functionals $theta''_1,ldots,theta''_{p-1}$ of $mathbb{C}^{N+1}$ such that the cardinality of $$lbrace theta''_1=cdots=theta''_{p-1}=0rbracecap iota''(X)$$ is equal to $d$. I would like to check that the functional $theta_i''$ above can be taken of the form $$theta_i''=theta_iotimestheta_i',$$ where $theta_i$ and $theta_i$' are linear functionals of $mathbb{C}^{n+1}$ and $mathbb{C}^{n'+1}$, respectively.
This a step (not proved) of Hoyt's proof of the topological invariance of the Chow monoid.
Any help is well accepted. Thanks in advance.
algebraic-geometry proof-writing
asked Dec 14 '18 at 8:16
Vincenzo ZaccaroVincenzo Zaccaro
1,254720
$endgroup$
Let $X$ be an irreducible complex projective variety of dimension $p$ and consider two projective embeddings $iotacolon Xsubseteq mathbb{P}^n$ and $iota'colon Xsubseteq mathbb{P}^{n'}$. Denote by $Sigmacolon mathbb{P}^ntimesmathbb{P}^{n'}longrightarrow mathbb{P}^N$ the Segre embedding. Then we get another embedding $iota''colon Xsubseteqmathbb{P}^N$ given by $$xlongmapsto (iota(x),iota'(x))longmapstoSigmabig((iota(x),iota'(x)).$$ Assume that that $iota''colon Xsubseteqmathbb P^N$ has degree $d$. Then there exist $(p-1)$ linear functionals $theta''_1,ldots,theta''_{p-1}$ of $mathbb{C}^{N+1}$ such that the cardinality of $$lbrace theta''_1=cdots=theta''_{p-1}=0rbracecap iota''(X)$$ is equal to $d$. I would like to check that the functional $theta_i''$ above can be taken of the form $$theta_i''=theta_iotimestheta_i',$$ where $theta_i$ and $theta_i$' are linear functionals of $mathbb{C}^{n+1}$ and $mathbb{C}^{n'+1}$, respectively.
This a step (not proved) of Hoyt's proof of the topological invariance of the Chow monoid.
Any help is well accepted. Thanks in advance.
algebraic-geometry proof-writing
algebraic-geometry proof-writing
asked Dec 14 '18 at 8:16
Vincenzo ZaccaroVincenzo Zaccaro
1,254720
asked Dec 14 '18 at 8:16
Vincenzo ZaccaroVincenzo Zaccaro
1,254720
asked Dec 14 '18 at 8:16
Vincenzo ZaccaroVincenzo Zaccaro
1,254720
asked Dec 14 '18 at 8:16
Vincenzo ZaccaroVincenzo Zaccaro
1,254720
asked Dec 14 '18 at 8:16
Vincenzo ZaccaroVincenzo Zaccaro
1,254720
1,254720
add a comment |
add a comment |
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