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By Alexander Grothendieck

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78. 3. NON-STABLE STRATA AND PLANE SEXTICS, I 37 (1) there exists W ∈ (ΘA \ {W }) with [w] ∈ W , or (2) dim(A ∩ Fw ∩ SW ) ≥ 2. 4. As is easily checked B(W, A) is a closed subset of P(W ). 10). Now let [v0 ] ∈ P(W ) be as above and let K := ρvV00 (A ∩ Fv0 ). 14) 2 W0 ) ⊂ P( 2 V0 / W0 ). 5. Keep notation as above. 15) W0 ) ∩ Gr(2, V0 )W0 = ∅. ) Proof. Let’s prove that [v0 ] ∈ B(W, A) if and only if (a) P(K) ∩ Gr(2, V0 ) is not equal to the singleton { 2 W0 }, or 2 (b) P(K) ∩ Θ 2 W0 Gr(2, V0 ) is not equal to the singleton { W0 }.

We have gω0 = (det gW )−1 ω0 because gω = ω. 26). 3. Non-stable strata and plane sextics, I In the present section we will prove the following result. 1. 1) 3 V ) and suppose that it belongs to BA ∪ BA∨ ∪ BC1 ∪ BC2 ∪ BE1∨ ∪ BE2∨ ∪ BF1 ∪ BF2 . Then there exists W ∈ ΘA such that CW,A is not a curve with simple singularities, more precisely either CW,A = P(W ) or else CW,A is a sextic curve and (1) there exists [v0 ] ∈ CW,A such that mult[v0 ] CW,A ≥ 4 if A ∈ BA , (2) CW,A is singular along a line (and hence non-reduced) if A ∈ (BC2 ∪ BE2∨ ∪ BF1 ∪ BF2 ), (3) CW,A is singular along a conic (and hence non-reduced) if A ∈ BE1∨ , (4) CW,A is singular along a cubic (and hence equal to a double cubic)) if A ∈ (BA∨ ∪ BC1 ).

The statement is equivalent to gW (P ) = (det gW )−2 P . Write A = 3 9 U W ⊕ B where B ∈ LG(EW ). Then ω = α ∧ ω0 where α ∈ W and ω0 ∈ B. We have gω0 = (det gW )−1 ω0 because gω = ω. 26). 3. Non-stable strata and plane sextics, I In the present section we will prove the following result. 1. 1) 3 V ) and suppose that it belongs to BA ∪ BA∨ ∪ BC1 ∪ BC2 ∪ BE1∨ ∪ BE2∨ ∪ BF1 ∪ BF2 . Then there exists W ∈ ΘA such that CW,A is not a curve with simple singularities, more precisely either CW,A = P(W ) or else CW,A is a sextic curve and (1) there exists [v0 ] ∈ CW,A such that mult[v0 ] CW,A ≥ 4 if A ∈ BA , (2) CW,A is singular along a line (and hence non-reduced) if A ∈ (BC2 ∪ BE2∨ ∪ BF1 ∪ BF2 ), (3) CW,A is singular along a conic (and hence non-reduced) if A ∈ BE1∨ , (4) CW,A is singular along a cubic (and hence equal to a double cubic)) if A ∈ (BA∨ ∪ BC1 ).

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