As an editor I often encounter the symbol $g_d^r$ as a noun. I tried googling but I only get papers where the symbol is used without a definition. Can someone supply a reference to a definition? Examples of usage: "Koszul cohomology groups of $g_d^r$s on singular nodal curves"; "any other basepoint-free $g_h^1$ [...] is composed with the given $g_d^1$.
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$\begingroup$ Sorry for the non-research question but I figure this is the one place where I'm sure to get an accurate answer. $\endgroup$– Silvio LevyCommented Oct 22, 2014 at 9:19
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$\begingroup$ I had the raw TeX in the title in order to facilitate searches. Is that not kosher? $\endgroup$– Silvio LevyCommented Oct 22, 2014 at 10:13
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As I understand, a $g_d^r$ is a linear system of dimeansion $r$ and degree $d$. Basically, these give you maps to $\mathbb{P}^r$ of degree $d$. The simplest example is of course hyperelliptic curves; these are the same as a $g^1_2$.
The reference that I have for this is Harsthorne, Algebraic Geometry, p. 341.
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$\begingroup$ And if Hartshorne used a notation, you can bet a lot of other people are using it too, after learning from him. $\endgroup$– David Roberts ♦Commented Oct 22, 2014 at 9:29
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2$\begingroup$ That is the notation for a linear system, as you say. However, these do not all give maps (there is the issue of base locus). Precisely, it is a pair of an invertible sheaf of degree $d$ and an $r+1$-dimensional subspace of the vector space of global sections. $\endgroup$ Commented Oct 22, 2014 at 9:30
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1$\begingroup$ @DavidRoberts: It is an old notation, far predating Hartshorne. $\endgroup$ Commented Oct 22, 2014 at 9:30
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$\begingroup$ Yeah, I wasn't really suggesting that his was the first usage, just the one that I have on hand. $\endgroup$ Commented Oct 22, 2014 at 9:32
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$\begingroup$ Perfect, thank you. (Re Jason's comment: Hartshorne calls the notation "classical" in the passage where he defines it.) $\endgroup$ Commented Oct 22, 2014 at 9:51