Pluricanonical specialization under a blowup: a clarification and repair of a proof step in A higher dimensional Hilbert irreducibility theorem

Blowing up one point of a constant family of smooth projective curves produces a reduced nodal fiber whose pluricanonical section space exceeds the fiber of the pluricanonical direct image by exactly m-1 in tensor degree m >= 2. We identify the specialization image and prove that no map from the full contracted special-fiber pushforward to the original pluricanonical sheaf extends the identity away from the contraction point. This gives an explicit obstruction to a general specialization inference appearing in Bresciani's positivity argument. The positivity conclusion is nevertheless recovered: the birational pluri-trace construction of Dutta-Murayama supplies the required total-space map, and an equivariant restriction argument avoids the full special-fiber identification. We verify the extension, generic injectivity, and equivariance, obtaining a repaired proof of the cited positivity proposition. The blowup facts and the pluri-trace construction are established material; the purpose is a focused clarification and repair, not a counterexample to the positivity or arithmetic conclusions. The upload contains the manuscript PDF and its LaTeX source archive. AI assistance: OpenAI ChatGPT assisted with English editing, mathematical checking, and organization. The author is responsible for the final text and its mathematical claims.