证明
Lemma 12 ([The row subscheme]). For each X∈ F_C , the scheme Z'_X is finite of…Lemma 13. Let C⊂^2_K be a reduced curve over an algebraically closed field K ,…观点
Fix a curve C in the image and its fiber F_C . Form I_C= (X,Z')∈ F_C^N(C): R_X|_ Z' =0 . This is a closed incidence. Indeed, restrict the row entries to the universal finite flat subscheme over the Hilbert scheme and push forward their common twist. This produces sections of a vector bundle on the product, whose zero locus is I_C . Every matrix X occurs in the incidence with its subscheme Z'_X . Conversely, row vanishing on Z' means J'_X⊂I_ Z' , so Z'⊂Z'_X . The scheme Z' has length N by its membership in the Hilbert scheme, and Z'_X has length N by Lemma 12 ([The row subscheme]). Therefore they are equal. The same lemma shows that each fiber of I_C over Hilb^N(C) has at most one geometric point. Such a finite-type fiber has dimension zero when nonempty, even if it is nonreduced. Projection to F_C is surjective on geometric points, so F_C ≤ I_C ≤^N(C) ≤N, where the last inequality is Lemma 13.