Lemma 11. Let K be an algebraically closed field, and let k≥1 and T≥0 be integers. In the degree- k plane linear system over K , the locus of reduced curves satisfying δ(C)≥T is constructible. If nonempty, it has dimension at most …

Write C=C_1∪⋯∪C_a , where the integral components have degrees b_i and normalization genera g_i . The Euler characteristics of the normalization and the curve are a-∑_i g_i and 1-p_a(C) . The normalization exact sequence therefore gives δ(C)=p_a(C)-∑_ i=1 ^a g_i+a-1, p_a(C)=(k-1)(k-2)2. In particular, the term a-1 accounts for the disconnected normalization, and the identity includes the contribution of intersections between different components. For fixed degrees and genera, multiplying the equations of the components defines a morphism from the product of their integral curve loci to the degree- k linear system. Restricting to distinct factors gives precisely the reduced curves of this type. The image is constructible, and its dimension is at most ∑_ i=1 ^a(3b_i+g_i-1) , =3k+∑_ i=1 ^a g_i-a; , =k(k+3)2-δ(C). There are only finitely many degree and genus types: a≤k , ∑b_i=k , and 0≤g_i≤(b_i-1)(b_i-2)/2 . The normalization identity determines δ(C) from each type. The required locus is therefore a finite union of constructible images with the stated dimension bound.