Subschemes of projective varieties

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I'm studying the article Introduction to Lawson Homology of Peters and Kosarew. In their exposition of Lawson homology they call projective any zero-locus $X$ of homogeneous polynomials in the complex projective space and, in particular, they consider Chow varieties $mathcalC_p,d(X)$ only for varieties of this type.



At page 8 they give the following definition-lemma.



enter image description here



Then they say:



enter image description here



From this one see that $Zin mathcalC_m+t,d(Ttimes X)$ and this leads to my question:



Question: Can one see a subscheme of a projective variety as a cycle?




algebraic-geometry reference-request




share|cite|improve this question



















  • It looks like by definition-Lemma 7 that subschemes of $X$ are cycles if you take $T$ to be a point. Is that what you want?
    – Samir Canning
    Aug 7 at 23:52










  • Nope. I mean something different. If Z is a subscheme is there a way to see it as $sum_in_iV_i$, where $n_iinmathbbN$ and $V_isubset Ttimes X$ is a irreducible projective algebraic set for any $i$?
    – Vincenzo Zaccaro
    Aug 7 at 23:59











  • I would say take the $V_i$ to be its irreducible components with the reduced scheme structure and the $n_i$ to be the length of $Z$ at the generic point of $V_i$.
    – Samir Canning
    Aug 8 at 0:04










  • Is $V_i$ the zero-locus of a homogeneous ideal?
    – Vincenzo Zaccaro
    Aug 8 at 0:07










  • As long as $Z$ is a closed subscheme (which is usually what people mean when they say subscheme) and you put the reduced scheme structure on them, it should be no problem. But then you could get the same cycle for lots of different schemes. I don’t know if that would be a feature or a bug.
    – Samir Canning
    Aug 8 at 0:19














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0
down vote

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I'm studying the article Introduction to Lawson Homology of Peters and Kosarew. In their exposition of Lawson homology they call projective any zero-locus $X$ of homogeneous polynomials in the complex projective space and, in particular, they consider Chow varieties $mathcalC_p,d(X)$ only for varieties of this type.



At page 8 they give the following definition-lemma.



enter image description here



Then they say:



enter image description here



From this one see that $Zin mathcalC_m+t,d(Ttimes X)$ and this leads to my question:



Question: Can one see a subscheme of a projective variety as a cycle?




algebraic-geometry reference-request




share|cite|improve this question



















  • It looks like by definition-Lemma 7 that subschemes of $X$ are cycles if you take $T$ to be a point. Is that what you want?
    – Samir Canning
    Aug 7 at 23:52










  • Nope. I mean something different. If Z is a subscheme is there a way to see it as $sum_in_iV_i$, where $n_iinmathbbN$ and $V_isubset Ttimes X$ is a irreducible projective algebraic set for any $i$?
    – Vincenzo Zaccaro
    Aug 7 at 23:59











  • I would say take the $V_i$ to be its irreducible components with the reduced scheme structure and the $n_i$ to be the length of $Z$ at the generic point of $V_i$.
    – Samir Canning
    Aug 8 at 0:04










  • Is $V_i$ the zero-locus of a homogeneous ideal?
    – Vincenzo Zaccaro
    Aug 8 at 0:07










  • As long as $Z$ is a closed subscheme (which is usually what people mean when they say subscheme) and you put the reduced scheme structure on them, it should be no problem. But then you could get the same cycle for lots of different schemes. I don’t know if that would be a feature or a bug.
    – Samir Canning
    Aug 8 at 0:19












up vote
0
down vote

favorite









up vote
0
down vote

favorite











I'm studying the article Introduction to Lawson Homology of Peters and Kosarew. In their exposition of Lawson homology they call projective any zero-locus $X$ of homogeneous polynomials in the complex projective space and, in particular, they consider Chow varieties $mathcalC_p,d(X)$ only for varieties of this type.



At page 8 they give the following definition-lemma.



enter image description here



Then they say:



enter image description here



From this one see that $Zin mathcalC_m+t,d(Ttimes X)$ and this leads to my question:



Question: Can one see a subscheme of a projective variety as a cycle?




algebraic-geometry reference-request




share|cite|improve this question











I'm studying the article Introduction to Lawson Homology of Peters and Kosarew. In their exposition of Lawson homology they call projective any zero-locus $X$ of homogeneous polynomials in the complex projective space and, in particular, they consider Chow varieties $mathcalC_p,d(X)$ only for varieties of this type.



At page 8 they give the following definition-lemma.



enter image description here



Then they say:



enter image description here



From this one see that $Zin mathcalC_m+t,d(Ttimes X)$ and this leads to my question:



Question: Can one see a subscheme of a projective variety as a cycle?





algebraic-geometry reference-request





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Aug 7 at 18:05









Vincenzo Zaccaro

1,411619




1,411619











  • It looks like by definition-Lemma 7 that subschemes of $X$ are cycles if you take $T$ to be a point. Is that what you want?
    – Samir Canning
    Aug 7 at 23:52










  • Nope. I mean something different. If Z is a subscheme is there a way to see it as $sum_in_iV_i$, where $n_iinmathbbN$ and $V_isubset Ttimes X$ is a irreducible projective algebraic set for any $i$?
    – Vincenzo Zaccaro
    Aug 7 at 23:59











  • I would say take the $V_i$ to be its irreducible components with the reduced scheme structure and the $n_i$ to be the length of $Z$ at the generic point of $V_i$.
    – Samir Canning
    Aug 8 at 0:04










  • Is $V_i$ the zero-locus of a homogeneous ideal?
    – Vincenzo Zaccaro
    Aug 8 at 0:07










  • As long as $Z$ is a closed subscheme (which is usually what people mean when they say subscheme) and you put the reduced scheme structure on them, it should be no problem. But then you could get the same cycle for lots of different schemes. I don’t know if that would be a feature or a bug.
    – Samir Canning
    Aug 8 at 0:19
















  • It looks like by definition-Lemma 7 that subschemes of $X$ are cycles if you take $T$ to be a point. Is that what you want?
    – Samir Canning
    Aug 7 at 23:52










  • Nope. I mean something different. If Z is a subscheme is there a way to see it as $sum_in_iV_i$, where $n_iinmathbbN$ and $V_isubset Ttimes X$ is a irreducible projective algebraic set for any $i$?
    – Vincenzo Zaccaro
    Aug 7 at 23:59











  • I would say take the $V_i$ to be its irreducible components with the reduced scheme structure and the $n_i$ to be the length of $Z$ at the generic point of $V_i$.
    – Samir Canning
    Aug 8 at 0:04










  • Is $V_i$ the zero-locus of a homogeneous ideal?
    – Vincenzo Zaccaro
    Aug 8 at 0:07










  • As long as $Z$ is a closed subscheme (which is usually what people mean when they say subscheme) and you put the reduced scheme structure on them, it should be no problem. But then you could get the same cycle for lots of different schemes. I don’t know if that would be a feature or a bug.
    – Samir Canning
    Aug 8 at 0:19















It looks like by definition-Lemma 7 that subschemes of $X$ are cycles if you take $T$ to be a point. Is that what you want?
– Samir Canning
Aug 7 at 23:52




It looks like by definition-Lemma 7 that subschemes of $X$ are cycles if you take $T$ to be a point. Is that what you want?
– Samir Canning
Aug 7 at 23:52












Nope. I mean something different. If Z is a subscheme is there a way to see it as $sum_in_iV_i$, where $n_iinmathbbN$ and $V_isubset Ttimes X$ is a irreducible projective algebraic set for any $i$?
– Vincenzo Zaccaro
Aug 7 at 23:59





Nope. I mean something different. If Z is a subscheme is there a way to see it as $sum_in_iV_i$, where $n_iinmathbbN$ and $V_isubset Ttimes X$ is a irreducible projective algebraic set for any $i$?
– Vincenzo Zaccaro
Aug 7 at 23:59













I would say take the $V_i$ to be its irreducible components with the reduced scheme structure and the $n_i$ to be the length of $Z$ at the generic point of $V_i$.
– Samir Canning
Aug 8 at 0:04




I would say take the $V_i$ to be its irreducible components with the reduced scheme structure and the $n_i$ to be the length of $Z$ at the generic point of $V_i$.
– Samir Canning
Aug 8 at 0:04












Is $V_i$ the zero-locus of a homogeneous ideal?
– Vincenzo Zaccaro
Aug 8 at 0:07




Is $V_i$ the zero-locus of a homogeneous ideal?
– Vincenzo Zaccaro
Aug 8 at 0:07












As long as $Z$ is a closed subscheme (which is usually what people mean when they say subscheme) and you put the reduced scheme structure on them, it should be no problem. But then you could get the same cycle for lots of different schemes. I don’t know if that would be a feature or a bug.
– Samir Canning
Aug 8 at 0:19




As long as $Z$ is a closed subscheme (which is usually what people mean when they say subscheme) and you put the reduced scheme structure on them, it should be no problem. But then you could get the same cycle for lots of different schemes. I don’t know if that would be a feature or a bug.
– Samir Canning
Aug 8 at 0:19















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