# Subschemes of projective varieties

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 $\mathcal{C}_{p,d}(X)$ only for varieties of this type.

At page 8 they give the following definition-lemma.

Then they say:

From this one see that $Z\in \mathcal{C}_{m+t,d}(T\times X)$ and this leads to my question:

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

• 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 '18 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 $n_i\in\mathbb{N}$ and $V_i\subset T\times X$ is a irreducible projective algebraic set for any $i$? – Vincenzo Zaccaro Aug 7 '18 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 '18 at 0:04
• Is $V_i$ the zero-locus of a homogeneous ideal? – Vincenzo Zaccaro Aug 8 '18 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 '18 at 0:19