In our scriptum we're talking about singularities. And there is the term "fat point" (for example of "tangent of fat point") . I cannot find any definition :-/ Has somebody an idea?
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7$\begingroup$ Usually means a point scheme with non-reduced structure sheaf. $\endgroup$– MohanCommented Jun 26, 2017 at 12:34
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$\begingroup$ What is a "scriptum"? $\endgroup$– Gerry MyersonCommented Sep 17, 2023 at 1:11
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2 Answers
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The fat point supported at $Z$ with multiplicity $k$ is the non-reduced scheme defined by $I(Z)^k$. This is often denoted $kZ$ in the fat points literature; it must not be confused with a zero-cycle. Slightly more generally, the fat point scheme supported at distinct points $Z_1,\dotsc,Z_t$ with multiplicities $k_1,\dotsc,k_t$ is the union of the fat points, that is the scheme defined by $\bigcap I(Z_i)^{k_i}$. This is often denoted $k_1 Z_1 + \dotsb + k_t Z_t$.
For a fat point scheme in affine or projective $n$-dimensional space, $\deg(kZ) = \binom{k+n-1}{n}$ and $\deg(k_1 Z_1 + \dotsb + k_t Z_t) = \sum \binom{k_i + n-1}{n}$ (not degree $k$ or $k_1+\dotsb+k_t$).
answered Jun 26, 2017 at 13:11
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2$\begingroup$ This only makes sense if $Z$ is a (reduced) closed point embedded in some ambient scheme. Moreover, it is by no means true that every fat point is of this form! $\endgroup$ Commented Jun 26, 2017 at 22:50
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$\begingroup$ Well, people use terminology to mean whatever they want it to mean. The people who actually study fat points use the definition that I've given (where, yes, $Z$ is a reduced point). $\endgroup$ Commented Jun 26, 2017 at 22:54
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$\begingroup$ And yes, embedded, as you say. $\endgroup$ Commented Jun 26, 2017 at 23:15
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$\begingroup$ Ok, but my other comment still stands: a point in $\mathbb A^n$ can have many other scheme structures than the ones you list, and I personally think those equally well deserve to be called 'fat/thick points'. $\endgroup$ Commented Jun 27, 2017 at 1:41
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1$\begingroup$ Although a review of the literature does seem to suggest that some modern texts reserve 'fat point' to mean specifically the $n$-th infinitesimal neighbourhood of a point embedded into a smooth $k$-variety, as opposed to more general 'thick points' that are arbitrary Artinian schemes. $\endgroup$ Commented Jun 27, 2017 at 2:02
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The definition I know is definition 1A.2 in "Lecture Notes on Motivic Cohomology" by Mazza, Voevodsky, Weibel.
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2$\begingroup$ Warning: this is a completely unusual definition. The standard one is the one discussed in the comments.. $\endgroup$– abxCommented Sep 16, 2023 at 14:42
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$\begingroup$ Thank you very much for your comments. I will try my best to see that these definitions are equivalent. $\endgroup$ Commented Sep 16, 2023 at 17:24