Wikipedia:Reference desk/Archives/Mathematics/2023 October 12
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October 12
[edit]algebraic vs. projective
[edit]I am looking for a reference related to Complex algebraic variety and the reason is that the figure in the article is too comprehensive and I am confused whether the topic of the article is algebraic or (vs) projective. It may be related to improving the article, but I thought the search for references was insufficient, so I posted it here. SilverMatsu (talk) 03:09, 12 October 2023 (UTC)
- The article Riemann sphere explains how the Riemann sphere can be visualized as the complex number plane wrapped around a sphere by a stereographic projection. If you know this fact and are familiar with the concept of stereographic projection, the figure used in Complex algebraic variety becomes clear. As it is now, without explanation, it is indeed puzzling to those not yet in the know. --Lambiam 04:18, 12 October 2023 (UTC)
- Thank you for your reply. I agree that the figure is too technical in the lead sentence. I think the wiki-links you provided is very helpful in explaining the figure. Perhaps algebraic curves will also be useful. Also, I'm thinking of adding Algebraic Curves and Riemann Surfaces by Rick Miranda as a reference. However, it seems necessary to keep in mind that the title of the article is not complex algebraic curves, but complex algebraic varieties. --SilverMatsu (talk) 07:01, 13 October 2023 (UTC)
- I've replaced the image, which is in a sense equally technical but more inviting to click the link "Riemann sphere" in the caption. I'm not familiar with the book; does it support specific statements in the article? Complex algebraic curves are one-dimensional complex algebraic varieties. Embeddings of Riemann surfaces in Euclidean or projective space are in general not algebraic and only two-dimensional, so the connection may be tenuous. --Lambiam 12:03, 13 October 2023 (UTC)
- I agree with replacing the image, I think reading the article Riemann sphere as a substitute for adding an explanation to the lead sentence. Also, I found two lecture notes: Some notes on algebraic curves and Riemann surfaces by Robbin (pdf), Modular Functions and Modular Forms by Milne (§. Riemann surfaces as algebraic, Proposition 2.21.) --SilverMatsu (talk) 15:50, 14 October 2023 (UTC)
- I've replaced the image, which is in a sense equally technical but more inviting to click the link "Riemann sphere" in the caption. I'm not familiar with the book; does it support specific statements in the article? Complex algebraic curves are one-dimensional complex algebraic varieties. Embeddings of Riemann surfaces in Euclidean or projective space are in general not algebraic and only two-dimensional, so the connection may be tenuous. --Lambiam 12:03, 13 October 2023 (UTC)
- Thank you for your reply. I agree that the figure is too technical in the lead sentence. I think the wiki-links you provided is very helpful in explaining the figure. Perhaps algebraic curves will also be useful. Also, I'm thinking of adding Algebraic Curves and Riemann Surfaces by Rick Miranda as a reference. However, it seems necessary to keep in mind that the title of the article is not complex algebraic curves, but complex algebraic varieties. --SilverMatsu (talk) 07:01, 13 October 2023 (UTC)