Unique lc-place residue pushforward and subadjoint discriminant effectivity

Normality of X gives ${\mu }_{\ast }{O}_Y(P)=O_X$. A rational function regular away from the exceptional locus with poles bounded by the effective exceptional P has no pole at any codimension-one point of X and hence is regular on X. Conversely regular functions pull back regularly. This proves the equality as sheaves, not just as global sections.

Because E is a Cartier divisor on smooth Y, there is an exact sequence $0\to O_Y(P-E)\to O_Y(P)\to O_E(P)\to 0$.

Pushforward and the assumed R¹ vanishing give $0\to I_W\to O_X\to {\mu }_{\ast }{O}_E(P)\to 0$.

The quotient is $O_W$. The image of 1 under the quotient identification is precisely the restriction to E of the canonical section of $O_Y(P)$. Thus this identifies the relevant map as well as the abstract quotient sheaf.

Since P is effective and does not contain E, multiplication by its restricted canonical section gives an injection $O_E\to O_E(P)$. Write $g=\mu \mathrm{\mid }E$. Pullback of regular functions gives an injection $O_W\to g_{\ast }{O}_E$, because W and E are integral and g is dominant. The composite $O_W\to g_{\ast }{O}_E\to g_{\ast }{O}_E(P)=O_W$

is the identity by the explicit image-of-1 description in Section 1. Since the second arrow is injective, its image lies in $O_W$ and contains all of $O_W$, so $g_{\ast }{O}_E=O_W$ canonically.

Here is the direct normality argument. Let U⊂W be affine and h∈K(W) integral over $O_W(U)$. Pull h back to the function field of E. At every point of $g^{-1}U$, the same monic equation makes it integral over the local normal ring of smooth E; hence it is regular there. Therefore it is a section of $g_{\ast }{O}_E(U)=O_W(U)$. Every such affine coordinate ring is integrally closed, so W is normal. No normalization or connectedness theorem is required for this deduction.

Let $K=K(W)$ and η its generic point. Localization of the direct-image equality gives $H^0(E_{\eta },O)=K$. This localization can be checked on a finite affine Čech cover of the proper separated generic fibre: localization commutes with the finite kernel computing H⁰. The same flat Čech argument after a field extension gives $H^0(E_{\eta }{\times }_{K}Kbar,O)=Kbar$.

A disconnected scheme has a nontrivial idempotent global function, obtained by setting 0 and 1 on the two clopen parts. The displayed field has none, proving geometric generic connectedness. The generic fibre is regular, since its affine rings are localizations of the regular rings of Y restricted to smooth E; over the characteristic-zero field K, a regular finite-type scheme is geometrically regular. Thus it is a smooth geometrically connected generic fibre. In the codimension-two case it is a curve; smooth connected components of a smooth curve are disjoint, so it is geometrically integral.

This is the precise connectedness used to reconstruct the generic marked rational curve. Connectedness of EVERY special fibre would require a separate Stein-factorization/formal-functions bridge; it is not claimed here or used in this component. The stable rational family, once separately constructed, has connected special fibres by its own definition and construction.

Fix Q as in the Statement. Near its generic image in the now normal W, ν is an isomorphism and Q has a uniformizing rational function t. Suppose for contradiction that $d_j<1-w_j$ for every component over Q. Since $w_j$ is a positive integer, $\lceil -d_j\rceil \ge w_j$

for every j. The rounding compatibility gives $P\mathrm{\mid }E\ge f^{\ast }Q$ over that generic point. After removing the images of the finitely many other boundary divisors, the rational function $1\mathrm{/}{g}^{\ast }t$ has divisor at least −P|E and so is a local section of $O_E(P)$. By $g_{\ast }{O}_E(P)=O_W$ it must be regular at the generic point of Q downstairs. But 1/t has a simple pole in that discrete valuation ring, a contradiction.

Hence some $d_j\ge 1-w_j$. For that j, $(d_j+w_j-1)\mathrm{/}{w}_j\ge 0$, so the maximum ${\delta }_Q$ is nonnegative. This proves exactly the needed effectivity after pushing the discriminant to W; exceptional Q do not contribute to that pushforward and are not constrained by this argument. Coefficients $d_j=0$ are included, so a reduced unmarked branch $w_j=1$ already satisfies the required inequality.