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I am very new to algebraic geometry. I was reading about divisors on a scheme. I am wondering does there is some connection between the followings.

An elliptic curve over the complex plane we can think as a torus. Now assume we have a graph embedded in the torus. Then any cycle on the graph is a codimension 1 closed subscheme, that is a divisor. Now can we relate a Hamiltonian cycle to certain property of the divisor?

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    $\begingroup$ If you are thinking of an elliptic curve as a torus then you are working over the complex numbers (as you say). So you should consider complex codimension one subvarieties, which are finite unions of points (rather than cycles on a graph). $\endgroup$
    – Yosemite Stan
    Commented Apr 29 at 17:19
  • $\begingroup$ @Yosemite Stan Ok I see. So can we think cycles of graphs as some algebraic cycle of the variety? $\endgroup$
    – KAK
    Commented Apr 29 at 17:27
  • $\begingroup$ No it’s not an algebraic cycle. Is this something you heard? Maybe you could provide more content of what you’re thinking of. $\endgroup$
    – Yosemite Stan
    Commented Apr 29 at 17:40
  • $\begingroup$ @YosemiteStan No I have not heard it anywhere, just guessing. I want to know is there any way to relate the cycles of the graphs and some combinatorial property of them like length or hamiltonicity with some geometric objects of the varity and their geometric properties? More generally can we study the combinatorial properties of the graph via some algebro-geometric language? $\endgroup$
    – KAK
    Commented Apr 29 at 18:03

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