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I am looking for a proof for duality principle in projective geometry. There is an axiomatic and then a homogeneous coordinates development of projective geometry where dual of lines to points and vise versa are defined and dual curves as well. Is there a proof of this duality within that context or does it have to be based on axiomatic development?

Edit : the axiomatic treatment even if proved as theorems does not include “curves”. How do we prove after defining dual curves that the duality principle extends to statements which include curves ? The comment leaves this unanswered.

Edit : The duality principle is about proof of theorems containing points and lines whereby a new theorem can be proven by interchanging points and lines and dualizing the statements but typically not containing curves. But by showing the duality for the curves namely that the dual of dual is the original one, we can extend it to that case.proof of extension ?

alireza
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    Usually any other development of projective geometry proves the standard axioms as theorems, so you can still still prove duality. – Peter Mar 26 '21 at 07:22
  • @Peter I have seen the proof that the dual of a line is a point and a point a line and curve dual in a curve in dual space. But these to me appear insufficient to prove duality, are they? – alireza Mar 26 '21 at 07:55
  • @Peter the book (Ueno) I am reading says now that we have proven the dual of a dual of a curve is the curve itself, the duality is complete. The book does not develop or prove the axioms and also the duality he is talking about includes curves not just lines and points. – alireza Mar 26 '21 at 08:06
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    What exactly do you mean by "duality principle"? Could you state it a bit more formally? – red_trumpet Mar 26 '21 at 09:41
  • General dual varieties (like dual curves, but also in higher dimensions) are discussed in §2 of Lamotke's paper The topology of complex projective varieties after S. Lefschetz – red_trumpet Mar 26 '21 at 09:44
  • From a parametrization of a curve, you can check that the dual curve of the dual curve is the original curve. – Jan-Magnus Økland Mar 26 '21 at 12:10
  • The duality principle is about proof of theorems containing point s and lines whereby a new theorem can be proven by interchanging points and lines and dualizing the statements but typically not containing curves. But by showing the duality for the curves namely that the dual of dual is the original one, we can extend it to that case.proof of extension ? – alireza Mar 27 '21 at 13:29

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