NONVARYING, AFFINE, AND EXTREMAL GEOMETRY OF STRATA OF DIFFERENTIALS

The paper proves Theorem 1.5: for k=1,2 and signatures μ with mi≥1 (abelian) or mi≥−1 with at least one of m1,m2>0 (quadratic), the divisor P^k_g{μ′} (merging the first two zeros) is an extremal effective divisor in P^k_g{μ}. The proof uses a global divisor-class relation 12λ − Dh − κμ η being a positive multiple of P^k_g{μ′} modulo other boundary components, and then intersects with Teichmüller curves C⊂P^k_g{μ′} to show C·P^k_g{μ′}/(C·η)=−1/(m1+m2+k)<0, concluding extremality via the argument of Gheorghita (all explicitly stated for k=1 and analogously for k=2) . The model’s solution gives a different, local proof: it works on the labeled cover, computes (m1+k)ψ1−η = m2·D12 + (other boundary) via a zero-splitting model, restricts to D12 to get D12·C = −ψ*·C, and then uses Teichmüller curves in the merged stratum to obtain D12·C<0 and hence extremality by a standard criterion. This matches the paper’s conclusion. The only issue is a sign/notation slip: the model defines η as c1(O(1)) but then uses the tautological relation η=(mi+k)ψi, which in the paper is stated for η=c1(O(−1)) ; interpreting η as c1(O(−1)) resolves the discrepancy without affecting the argument. Overall, both are correct, with different proofs.