Timeline for Why "monoidal" transformation?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Aug 14, 2019 at 11:11 | answer | added | Jonny Evans | timeline score: 10 | |
| Aug 14, 2019 at 8:33 | comment | added | Francesco Polizzi | @JonnyEvans: nice! If you find the time to write an answer about this, it would be greatly appreciated. | |
| Aug 14, 2019 at 8:18 | comment | added | Jonny Evans | Semple and Roth "Introduction to Algebraic Geometry" in the section "Examples on Chapter VIII", Example 12 discuss a characterisation of monoidal transformations in terms of monoids in the sense of Cayley. If I have time later, I'll try and translate what they're doing into something I understand. | |
| Aug 14, 2019 at 2:17 | comment | added | roy smith | I do not see any connection with the usage of Cayley, but Zariski defines a monoidal, and a quadratic transform on page 535 of his paper Foundations of a general theory of birational correspondences, TAMS, vol. 53, (1943) pp. 490-542. The distinction is apparently that a quadratic correspondence blows up a point, hence modulo normalization, is the only one needed for desingularizing a surface, while a monoidal transform blows up a subvariety. The subsequent paper Reduction of the singularities of algebraic three dim'l varieties, hence explicitly uses both terms, AMS vol 45, 1944, 472-542. | |
| Aug 14, 2019 at 1:24 | comment | added | Sándor Kovács | I always assumed that it was called "monoidal", because it came from the Cremona transform of $\mathbb P^2$, which is indeed monoidal: $[x:y:z]\mapsto [yz:xz:xy]$. I have no evidence to back this up. My thinking was that the Cremona tr. had to come first and they did indeed use that to partially resolve plane curves to have only nodes (and stayed inside $\mathbb P^2$ all the time). I further assumed that later on they (i.e., someone) realized that what is happening at three different points in the Cremona tr. simultaneously can be done individually if we are willing to leave $\mathbb P^2$... | |
| Aug 13, 2019 at 21:38 | comment | added | Lubin | My memory is not to be trusted, @FrancescoPolizzi, but I think I heard Zariski use the term “monoidal transformation” at a time when Hartshorne was still a very beginning graduate student. | |
| Aug 13, 2019 at 20:01 | history | edited | Francesco Polizzi | CC BY-SA 4.0 |
added 37 characters in body
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| Aug 13, 2019 at 19:58 | comment | added | Carlo Beenakker | my understanding for the use of the word "monoidal" by both Cayley and Hartshorne is indeed the Greek origin "monos". | |
| Aug 13, 2019 at 19:52 | history | edited | Francesco Polizzi | CC BY-SA 4.0 |
edited title
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| Aug 13, 2019 at 19:46 | comment | added | Francesco Polizzi | Anyway, I am pretty sure that the term predates Hartshorne (maybe it was already used by Zariski?) | |
| Aug 13, 2019 at 19:38 | comment | added | Francesco Polizzi | Does Hartshorne explain why it uses the term "monoidal"? Maybe because there is a single (=mono) point blown up? | |
| Aug 13, 2019 at 19:32 | comment | added | Carlo Beenakker | R. Hartshorne (1977) explains his introduction of the term as follows: We define a monoidal transformation of a surface to be the operation of blowing up a single point. This new terminology is to distinguish it from the more general process of blowing up an arbitrary closed subscheme. It also goes by many other names in the literature: locally quadratic transformation, dilatation, $\sigma$-process, Hopf map, to mention a few. It would seem Hartshorne was not aware of Cayley's terminology, or at least he does not refer to it. | |
| Aug 13, 2019 at 18:46 | history | asked | Francesco Polizzi | CC BY-SA 4.0 |