| import Mathlib |
| import OAI.AlgebraicGeometry.Seshadri.Intersection.AllCurveDegree |
|
|
| namespace OAI |
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|
| section |
| noncomputable section |
| |
| section |
|
|
| namespace MaximalSeshadri.Geometry |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
| open MaximalSeshadri.Frames MaximalSeshadri.Projective |
|
|
| theorem IntegralCurve.tensor_euler_add (S : Surface) (C : IntegralCurve S) |
| (A : LineBundle S.scheme) (hA : A.IsAmple) (L M : LineBundle C.scheme) : |
| eulerCharacteristic (C.embedding ≫ S.structureMap) 1 (L.tensor M).sheaf - |
| eulerCharacteristic (C.embedding ≫ S.structureMap) 1 M.sheaf = |
| eulerCharacteristic (C.embedding ≫ S.structureMap) 1 L.sheaf - |
| eulerCharacteristic (C.embedding ≫ S.structureMap) 1 (O C.scheme) := by |
| obtain ⟨σ,hσ,B,a,ha,hc⟩ := C.exists_projective_sections S A hA |
| let := hσ |
| let := hc |
| exact projective_curve_tensor_euler_add (C.embedding ≫ S.structureMap) |
| C.dimension a ha L M |
|
|
| theorem IntegralCurve.power_euler (S : Surface) (C : IntegralCurve S) |
| (A : LineBundle S.scheme) (hA : A.IsAmple) (L : LineBundle C.scheme) (n : ℕ) : |
| eulerCharacteristic (C.embedding ≫ S.structureMap) 1 (L.pow n).sheaf - |
| eulerCharacteristic (C.embedding ≫ S.structureMap) 1 (O C.scheme) = |
| (n : ℤ) * (eulerCharacteristic (C.embedding ≫ S.structureMap) 1 L.sheaf - |
| eulerCharacteristic (C.embedding ≫ S.structureMap) 1 (O C.scheme)) := by |
| obtain ⟨σ,hσ,B,a,ha,hc⟩ := C.exists_projective_sections S A hA |
| let := hσ |
| let := hc |
| exact projective_curve_power_euler (C.embedding ≫ S.structureMap) |
| C.dimension a ha L n |
|
|
| theorem IntegralCurve.pullback_power_euler (S : Surface) (C : IntegralCurve S) |
| (A : LineBundle S.scheme) (hA : A.IsAmple) (L : LineBundle S.scheme) (n : ℕ) : |
| eulerCharacteristic (C.embedding ≫ S.structureMap) 1 |
| ((L.pullback C.embedding).pow n).sheaf - |
| eulerCharacteristic (C.embedding ≫ S.structureMap) 1 (O C.scheme) = |
| (n : ℤ) * curveDegree S L C := |
| C.power_euler S A hA (L.pullback C.embedding) n |
|
|
| end |
| end MaximalSeshadri.Geometry |
| end |
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| end |
| end |
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| end OAI |
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|
