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In Carlo Beenakker's answer to this recent MO question, it turns out that the name "monoid" was first used in mathematics by Arthur Cayley for a surface of order $𝑛$ which has a multiple point of order $𝑛−1$.

On the other hand, a (old) name for the blow-up is "monoidal transformation". I suspect that there should be a link relating these two classical terminologies, but I was not able to find one. So let me ask the following

Question. Why was the blow-up classically known as "monoidal transformation"? Was this related to Cayley's "monoid"?

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    $\begingroup$ 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. $\endgroup$
    – Carlo Beenakker
    Commented Aug 13, 2019 at 19:32
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    $\begingroup$ 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. $\endgroup$
    – Lubin
    Commented Aug 13, 2019 at 21:38
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    $\begingroup$ 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 $\mathbb P^2$... $\endgroup$
    – Sándor Kovács
    Commented Aug 14, 2019 at 1:24
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    $\begingroup$ 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. $\endgroup$
    – roy smith
    Commented Aug 14, 2019 at 2:17
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    $\begingroup$ 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. $\endgroup$
    – Jonny Evans
    Commented Aug 14, 2019 at 8:18

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As I mentioned in the comments, Semple and Roth (Introduction to Algebraic Geometry) discuss this in "Examples on Chapter VIII" example 12 (p.187 of my version). As far as I can tell, this is what they are saying:

Homoloidal webs:

Let $V$ be a 4-dimensional vector space and suppose we're working in the projective 3-space $PV$ with homogeneous coordinates $[x_0:\cdots:x_3]$. Pick four homogeneous (degree $d$) polynomials $F_0,\ldots,F_3\in Sym^d(V)$. We get a rational map $PV\to PV$ defined by

$$[x_0:\cdots:x_3]\mapsto[F_0(x):\cdots:F_3(x)]$$

How should we think of this? We have a 3-parameter family of surfaces $\sum_i\lambda_iF_i=0$ (parametrised by $[\lambda_0:\cdots:\lambda_3]\in PV^*$, called a "homoloidal web" of surfaces) and the equation $\sum_i\lambda_iF_i(x)=0$ defines a hyperplane in $PV^*$ consisting of surfaces which contain $x$. Our map sends $x$ to the point in $PV$ dual to this hyperplane.

Monoids:

Now suppose that our surfaces $F_i$ all have a singularity of order $d-1$ at the point $p$. In other words, they're all monoids in the sense of Cayley (because they have degree $d$). They call the corresponding birational maps monoidal transformations. Certainly $p$ will be in the base locus of this system of surfaces, so you need to blow up at $p$ to get a morphism, but I'm still not sure why this is a reasonable definition.

They do discuss examples of this immediately afterwards, but I get lost in the notation.

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    $\begingroup$ Thanks. A remark: a "homaloidal" web means a web that defines a birational map onto $PV^*$, i.e. such that any three members have only one variable intersection. $\endgroup$
    – Francesco Polizzi
    Commented Aug 14, 2019 at 11:21

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