claimroutineinferred_dep

Ideal order along a branch equals weighted contact

Shows II has order RhPRh_P along a branch PP, the weighted contact
links the jet ideal to the contact heights in the curve inequality
Claim·§2.3 From the curve inequality to all coefficient packets·inside thmInterpolation with separated weights

Statement (paper)

For a branch PP through a center, with z0=tz_0=t and zi=ui′z_i=u_i', the ideal II satisfies ord⁡PI=min⁡a{(R/va)ord⁡Pza}=RhP\operatorname{ord}_P I=\min_a\{(R/v_a)\operatorname{ord}_P z_a\}=Rh_P. This uses that replacing log⁡(1+t)\log(1+t) by the Taylor polynomial Gi(t)G_i(t) preserves hPh_P, and that t=Y−1t=Y-1, ui′=Xi−cji−Gi(t)u_i'=X_i-c_{ji}-G_i(t) are regular local coordinates at aja_j.

In plain English

For a branch PP through a center, with local coordinates z0=tz_0=t and zi=ui′z_i=u_i', the ideal II has order along PP equal to min⁡a{(R/va)ord⁡Pza}\min_a\{(R/v_a)\operatorname{ord}_P z_a\}, which equals RhPRh_P.

Why it's here

This claim translates the weighted-jet ideal II into the branch-contact quantity hPh_P, which is the quantity the curve inequality controls. The key points are that swapping log⁡(1+t)\log(1+t) for its Taylor polynomial Gi(t)G_i(t) does not change hPh_P, and that tt and ui′u_i' are regular local coordinates at aja_j. This lets later steps, such as the nefness of A−(1+σ)EA-(1+\sigma)E on the blow-up, be checked curve by curve.
Classified routine:
The order of an ideal along a branch is the minimum of the orders of its generators. Here the coordinate change is regular, and the Taylor-truncation error has sufficiently high order not to change the weighted contact, using the earlier separated-weight choice.

Uses

claimSeparated weights avoid center dependence

Used by

claimCurve inequality makes blowup class nefclaimVanishing and projective sections generate jets

Sources

PaperOpen
Section
§2.3 From the curve inequality to all coefficient packets
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sections/interpolation.tex:457
Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

  • info
    inferred_dep
    outline: "Algebraic ideals encode weighted branch contact" rests on claim-geom-interpolation-1