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Stream: deprecated: algebraic geometry

Topic: homological equivalence of curves

Karem (May 07 2022 at 02:33):

Hi

Karem (May 07 2022 at 04:48):

if I wish to discuss algebraic geometry argument can I post it here? Is there latex rendering here?

Karem (May 07 2022 at 04:49):

I am new to Zulip.

John Baez (May 07 2022 at 04:49):

Yes, just enter the title of your question in the box above while you are writing your comment. (Or one of us can do it later.)

Karem (May 07 2022 at 04:51):

@John Baez Thanks!

Karem (May 07 2022 at 04:53):

Question:

The setting is as follows let us say that $X = C_1 \times C_2 \times C_3 \times C_4$ and $D$ is a surface in $X$. Assume I have a rational function on $D$ and $K$ being a curve in $X$ that is one of the components of the divisors of f. That is if $div(f)_D = \sum n_iK_i$ then $K := K_i$.

Karem (May 07 2022 at 04:53):

Consider the following surfaces $Z = Pr_{1,2,3}(K) \times C_4$ and $Z' = C_1 \times Pr_{2,3,4}(K)$. $K$ is subset of both of them.

I wish to construct curves $W_1,\ldots,W_m$ on surfaces $$Z$ and $Z'$$ in either one of surfaces or both where $W{i} \sim_{hom} W{i + 1}$ such that $mK - (W_1 - W_2) + \ldots (W_{m - 1} - W_m)$ is either zero or homologically equivalent to curves of the form $p \times V$ for $V$ a curve and p a point, where $W_i$ could also be $K$.

Karem (May 07 2022 at 04:54):

For instance suppose that $W_1 - K ~_{hom} \sum p_i \times V$ on $Z$ and $W_2 - W_1 ~_{hom} sum p_i \times V$ on $Z'$ such that $W_2 + K \sim_{hom} p_i \times V$ it follows $2K + (W_1 - K) + (W_2 - W_1) = K + W_2 \sim_{hom} \sum p_i \times V$.

John Baez (May 07 2022 at 04:54):

You need to use double dollar signs, like S$, to get LaTeX to work here.

Karem (May 07 2022 at 04:54):

oh okay thanks for letting me know I will update the above

John Baez (May 07 2022 at 04:54):

You can edit your question replacing each $ with SS.

Karem (May 07 2022 at 04:57):

$[K] = (a_1,b_1,v_1)$ in cohomology. If $v_1 = 0$ then we are done because then $K$ would be homologically equivalent to curves of the form $Pr_{1,2,3}(K) \times p + q \times C_4$ so for instance $K - (K - Pr_{1,2,3}(K) \times p + q \times C_4) = Pr_{1,2,3}(K) \times p + q \times C_4$

Karem (May 07 2022 at 04:57):

Pick an element $a$ in the first element of kunneth decomposition that is in the $Hdg^2(Pr_{1,2,3}(K) \times C_4)$. It follows that the element $(a,0,-v)$ is in $Hdg^2(Pr_{1,2,3}(K) \times C_4)$. By Lefschetz (1,1) theorem we have that there exists a curve $W_1$ such that $[W_1] = (a,0,-v)$. It follows that $K + W_1 = (a_1,b_1,v) + (a,0,-v) = (a_1 + a,b_1,0)$ so it follows that
$K + W_1 \sim_{hom} \sum p_i \times V$ on $Z_1$$

Karem (May 07 2022 at 04:59):

doing the same idea for $Z^{\prime} = C_1 \times Pr_{2,3,4}(K)$ we get that we can construct curve $W_2$ such that $W_2 + K \sim_{hom} \sum p_i \times V$.

Notice $W_2 \subset C_1 \times Pr_{2,3,4}(K) \subset C_1 \times Pr_{2,3}(K) \times C_4$ and similarly for $W_1$.

From the relation $K + W_1 \sim_{hom} \sum p_i \times V$ and $K + W_2 \sim_{hom} p_i \times V$ it follows that $-W_1 - W_2 \sim_{hom} HVC$ in $C_1 \times Pr_{2,3}(K) \times C_4$. Can we construct a surface subset of $C_1 \times Pr_{2,3}(K) \times C_4$ where the relation $-(W_1 + W_2) \sim_{hom} \sum p_i \times V$ holds?

Karem (May 07 2022 at 05:00):

I think that we can simple take the union $C_1 \times Pr_{2,3,4}(K) \cup Pr_{1,2,3}(K) \times C_4$ and the relation above should hold?

Karem (May 07 2022 at 05:08):

Essentially to cut the story short: Suppose we have a curve $K$ in $C_1 \times C_2 \times C_3 \times C_4$. Consider the following two surfaces $Z_1 = Pr_{1,2,3}(K) \times C_4$ and $Z_2 = C_1 \times Pr_{2,3,4}(K)$ can we construct curves $W_{i,1}$ and $W_{i,2}$ in $Z_j$ where $W_{i + 1,j} - W{i - 1,j} \sim{hom} \sum p_i \times V_i$ in $Z_j$ such that $mK + (W_{i + 1,j} - W_{i - 1,j} - \sum p_i \times Vi) + \ldots \sim_{hom} \sum s_i \times V_i$ on either $Z_1$ or $Z_2$ or union of both or other surfaces of that type. I claim my argument above should do that, though I would love to discuss it with people !

Karem (May 07 2022 at 05:09):

I am off to bed I am sorry for late reply I will definitely discuss tomorrow with any reply :)

Karem (May 07 2022 at 05:09):

Good night everyone!

Karem (May 07 2022 at 17:21):

morning

John Baez (May 07 2022 at 17:28):

Personally this question is too technical for me... but there may be someone around who can handle it. Since this is a category theory forum, you'll probably get more responses if you ask questions that visibly involve category theory.

Karem (May 07 2022 at 17:44):

@John Baez Sorry for late reply I was eating food. Awesome I will be more active in category theory :). Thanks

Matteo Capucci (he/him) (May 10 2022 at 20:46):

(I'm also not that well-versed in AG, but let me suggest to cross-post on mathoverflow.net)

Karem (May 10 2022 at 23:36):

@Matteo Capucci (he/him) Thank you. I will try to do that.