Construction of singular surfaces over a finite field
نویسنده
چکیده
Fix a prime power q and integers m ≥ 2, σ > 0, x > 0, g ≥ 0, d ≥ mσ + 1 such that q ≥ (δ − 1)δ3, where δ := d3 + 3dx + 4x + 2g − 2. Let C ⊂ P be a smooth degree x curve defined over Fq such that h(P,IC(σ − 1)) = h(C,OC (σ − 2)) = 0 and pa(C) = g. Here we prove the existence of a degree d surface X ⊆ P3 defined over Fq, such that Sing(X) = C and X has ordinary multiplicity m along C, i.e. for every P ∈ C(F̄q) the tangent cone of X at P is reduced and it is the union of m distinct planes containing the tangent line of C at P . Mathematics Subject Classification: 14J25; 14J70; 14N05; 12E20; 14B05; 14B25
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