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In his answer here Qing Liu mentioned "the 'discriminant' of X which measures the defect of a functorial isomorphism which involves powers of the relative dualizing sheaf of X/R."

Could somebody give a reference for or explain this?

I ask because there is a natural definition of conductor of a proper morphism to a normal scheme (for example as in Serre's "Algebraic groups and class fields"), and some relationship between the conductor and discriminant, but the definitions of discriminants I have seen always seem situation-specific and ad-hoc.


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up vote 9 down vote accepted

The "magistral paper of Takeshi Saito" mentioned by Qing Liu is the following: 'Conductor, Discriminant, and the Noether formula of Arithmetic surfaces', Duke Math Journal, vol. 57, no. 1. See here if you have a subscription to Duke: The definition of the discriminant is at the top of the second page. Or see section three of Liu's notes which he referenced in his reply to which you provided a link.

But for what it's worth, here is a quick answer (basically copied from Saito's paper):

Let $T$ be a nice scheme (spectrum of a field is enough for us), and $g:Y\to T$ a relative curve (i.e. geometrically connected, proper, relative dimension one). When $Y$ is smooth over $T$, there is a functorial isomorphism $\Delta:\det Rg_*(\omega_{Y/T}^{\otimes 2})\to(\det Rg_*\omega_{Y/T})^{\otimes 13}$. This is due to Deligne (unpublished letter to Quillen....).

Now let $\mathcal{O}_K$ be a Henselian discrete valuation ring with algebraically closed residue field, put $S=\mbox{Spec}\mathcal{O}_K$, and let $X\to S$ be a relative curve which is regular. We have an invertible $\mathcal{O}_K$-module

$V=\mbox{Hom} (\det Rg_*(\omega_{X/S}^{\otimes 2}),(\det Rg_*\omega_{X/S})^{\otimes 13})$.

and Deligne's result, applied to the generic fibre of $X$, produces a natural element $\Delta\in V\otimes_{\mathcal{O}_K} K$. The discriminant of $X$ is defined to be the order of $\Delta$ (that is, the unique $d\in\mathbb{Z}$ such that $\mathcal{O}_K\Delta=\mathfrak{p}^d V$).

I do not know if there is a similar notion of discriminant for higher dimensional $S$-schemes.

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The generic fiber has to be assumed smooth in the 2nd part (so the first part can be applied there). Too bad that Deligne is an oracle here -- i.e., no published reference for the proof of the existence of the asserted general isomorphism $\Delta$. Since 13 = 1 + 12, this also nicely generalizes all of the "classical" stuff with 12's in the genus-1 case. – BCnrd May 5 '10 at 12:34
Whoops, yes, you are right, thank you. I was imagining that $K$ was perfect (i.e. characteristic zero), so that the regularity of $X$ would imply the smoothness of the generic fibre. – Matthew Morrow May 5 '10 at 12:52
Thanks Matthew, sadly I can't access the journal at the moment, but I can probably find it in the library. Thanks to Brian for saying where the 13 comes from. – David Holmes May 5 '10 at 14:20
@David: I didn't actually say where the 13 comes from; I just said where the 12 comes from for genus 1! I assume the 13 comes out from an argument inspired by analogy with formulas from algebraic surfaces, but the oracle status of the Deligne reference is an obstruction to saying more. I would like it if someone will explain where the 13 comes from in this general setting. – BCnrd May 5 '10 at 14:49
This comes from computations on the Hodge bundle on the moduli space of stable curves (Mumford: "Stability of projective varieties", Enseignement Mathématiques, 1977): 1+12=1-6n+6n^2 (n=2). – Qing Liu May 5 '10 at 22:44

It seems I don't have enough reputation to make comments, so I'll write a comment here instead: The "Lettre à Quillen" mentioned in Matthew's answer is now available on Deligne's webpage:

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