“Nice” definition of discriminant as alluded to in an answer of Qing Liu

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.

Thanks.

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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: projecteuclid.org/euclid.dmj/1077306852 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 This is due to Deligne (unpublished letter to Quillen....).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:

http://publications.ias.edu/sites/default/files/Lettre-a-Quillen.pdf

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