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The Canonical Map and the Canonical Ring of Algebraic Curves

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The Canonical Map and the Canonical Ring of Algebraic Curves The canonical map and the canonical ring of algebraic curves. Vicente Lorenzo Garc´ıa The canonical map and the canonical ring of algebraic curves. Introduction. LisMath Seminar. Basic tools. Examples. Max Vicente Lorenzo Garc´ıa Noether's Theorem. Enriques- Petri's Theorem. November 8th, 2017 Hyperelliptic case. 1 / 34 Outline The canonical map and the canonical ring of algebraic curves. Vicente Introduction. Lorenzo Garc´ıa Introduction. Basic tools. Basic tools. Examples. Examples. Max Noether's Theorem. Max Noether's Theorem. Enriques- Petri's Theorem. Hyperelliptic Enriques-Petri's Theorem. case. Hyperelliptic case. 2 / 34 Outline The canonical map and the canonical ring of algebraic curves. Vicente Introduction. Lorenzo Garc´ıa Introduction. Basic tools. Basic tools. Examples. Examples. Max Noether's Theorem. Max Noether's Theorem. Enriques- Petri's Theorem. Hyperelliptic Enriques-Petri's Theorem. case. Hyperelliptic case. 3 / 34 The canonical map and the canonical ring of algebraic curves. Vicente Lorenzo Garc´ıa Definition Introduction. We will use the word curve to mean a complete, nonsingular, one dimensional scheme C Basic tools. over the complex numbers C. Examples. Max Noether's Definition Theorem. The canonical sheaf ΩC of a curve C is the sheaf of sections of its cotangent bundle. Enriques- Petri's Theorem. Hyperelliptic case. 4 / 34 The canonical map and the canonical ring of algebraic curves. Vicente Theorem (Serre's Duality) Lorenzo Garc´ıa Let L be an invertible sheaf on a curve C. Then there is a perfect pairing 0 −1 1 Introduction. H (C; ΩC ⊗ L ) × H (C; L) ! K: Basic tools. 1 0 −1 Examples. In particular, h (C; L) = h (C; ΩC ⊗ L ). Max Noether's Theorem. Theorem (Riemann-Roch) Enriques- Let L be an invertible sheaf on a curve C of genus g. Then, Petri's Theorem. 0 0 −1 h (C; L) − h (C; ΩC ⊗ L ) = deg L + 1 − g: Hyperelliptic case. 5 / 34 The canonical map and the canonical ring of algebraic curves. Vicente Definition Lorenzo Garc´ıa An invertible sheaf L on C is said to be generated by global sections if for any point P 2 C there is a global section of L not vanishing at P . Introduction. 0 Basic tools. If L is an invertible sheaf generated by global sections, a basis s0; : : : ; sn of H (C; L) has no common zeros and induces a map to projective space: Examples. Max n Noether's C ! P ;P 7! (s0(P ): ::: : sn(P )): Theorem. Enriques- Petri's Theorem. Definition Hyperelliptic If the previous map is a closed embedding we say that L is very ample. case. 6 / 34 The canonical map and the canonical ring of algebraic curves. Vicente Lorenzo Theorem Garc´ıa Let L be an invertible sheaf on a curve C. Then: Introduction. i) L is generated by global sections if and only if for every P 2 C, Basic tools. 0 0 Examples. h (C; L(−P )) = h (C; L) − 1: Max Noether's ii) L is very ample if and only if for every P; Q 2 C, Theorem. Enriques- h0(C; L(−P − Q)) = h0(C; L) − 2: Petri's Theorem. Hyperelliptic case. 7 / 34 The canonical map and the canonical ring of algebraic curves. Vicente Lorenzo Garc´ıa Definition Introduction. 1 A curve C of genus g ≥ 2 is hyperelliptic if there exists a degree 2 morphism C ! P . Basic tools. Examples. Theorem Max Noether's The canonical sheaf ΩC is always generated by global sections. Moreover, it is very ample Theorem. if and only if C is not hyperelliptic. Enriques- Petri's Theorem. Hyperelliptic case. 8 / 34 The canonical map and the Let C be a non-hyperelliptic curve of genus g ≥ 3. Our aim is to study its canonical ring: canonical ring of M 0 ⊗n algebraic H (C; Ω ): curves. C n≥0 Vicente Lorenzo Garc´ıa To do so, we are going to consider the map: Introduction. M ' : S∗H0(C; Ω ) ! H0(C; Ω⊗n): Basic tools. C C n≥0 Examples. Max Noether's We are going to see that it is surjective and we are going to study its kernel I. Theorem. Enriques- Petri's Remark Theorem. If C is non-hyperelliptic, ΩC is very ample and it induces an embedding Hyperelliptic case. g−1 C,! P : M 0 ⊗n The homogeneous coordinate ring of C for this embedding is precisely H (C; ΩC ): n≥0 9 / 34 Outline The canonical map and the canonical ring of algebraic curves. Vicente Introduction. Lorenzo Garc´ıa Introduction. Basic tools. Basic tools. Examples. Examples. Max Noether's Theorem. Max Noether's Theorem. Enriques- Petri's Theorem. Hyperelliptic Enriques-Petri's Theorem. case. Hyperelliptic case. 10 / 34 The canonical map and the canonical Lemma ring of 0 0 algebraic Let L and M be invertible sheaves on a curve C such that H (C; L);H (C; M) 6= 0. curves. Let V be the image of the map Vicente Lorenzo H0(C; L) ⊗ H0(C; M) ! H0(C; L ⊗ M): Garc´ıa Introduction. Then 0 0 Basic tools. dim V ≥ h (C; L) + h (C; M) − 1: Examples. Max Noether's Theorem (Clifford) Theorem. Let L be an invertible sheaf on a curve C such that 0 ≤ deg L ≤ 2g − 2. Then Enriques- Petri's 0 Theorem. 2(h (C; L) − 1) ≤ deg L: Hyperelliptic case. Furthermore, equality holds if and only if (i) L = OC or L = ΩC . 1 (ii) C is hyperelliptic and L is 2 (deg L)-times the invertible sheaf induced by the unique linear system of dimension 1 and degree 2 on C. 11 / 34 The canonical map and the canonical ring of algebraic Lemma (Base point free pencil trick) curves. Vicente Let L and M be invertible sheaves on a curve C and s1 and s2 two global sections of L Lorenzo 0 Garc´ıa having no common zeros. If V is the subspace of H (C; L) generated by s1 and s2, then the kernel of the map Introduction. V ⊗ H0(C; M) ! H0(C; L ⊗ M) Basic tools. is isomorphic to H0(C; M ⊗ L−1). Examples. Max Noether's Lemma (Castelnuovo) Theorem. Enriques- Let F be a coherent sheaf on a curve C and L an invertible sheaf on C generated by Petri's global sections, such that H1(C; F ⊗ L−1) = 0. Then the map Theorem. Hyperelliptic H0(C; F) ⊗ H0(C; L) ! H0(C; F ⊗ L) case. is surjective. 12 / 34 Outline The canonical map and the canonical ring of algebraic curves. Vicente Introduction. Lorenzo Garc´ıa Introduction. Basic tools. Basic tools. Examples. Examples. Max Noether's Theorem. Max Noether's Theorem. Enriques- Petri's Theorem. Hyperelliptic Enriques-Petri's Theorem. case. Hyperelliptic case. 13 / 34 The canonical map and the Let C be a non-hyperelliptic curve of genus 3. canonical ring of algebraic By Riemann-Roch's Theorem, curves. 0 Vicente h (C; ΩC ) = 3, Lorenzo 0 ⊗n Garc´ıa h (C; ΩC ) = 4n − 2; 8n > 1. Introduction. Now, if P 2 C then, Basic tools. 0 0 0 H (C; ΩC (−2P )) ⊂ H (C; ΩC (−P )) ⊂ H (C; ΩC ) Examples. Max and Noether's 0 Theorem. h (C; ΩC (−2P )) = 1, 0 Enriques- h (C; ΩC (−P )) = 2, Petri's Theorem. 0 h (C; ΩC ) = 3, Hyperelliptic 0 case. so we can find a basis fr; s; tg of H (C; ΩC ) such that, ordP (r) = 2, ordP (s) = 1, ordP (t) = 0. 14 / 34 The canonical map and the canonical ring of algebraic Having said that, we consider the map curves. Vicente 0 0 0 ⊗2 Lorenzo H (C; ΩC (−P )) ⊗ H (C; ΩC ) ! H (C; ΩC (−P )) Garc´ıa Introduction. By the base point free pencil trick, it is surjective. Hence, Basic tools. 0 ⊗2 2 2 Examples. H (C; ΩC (−P )) = hr ; rs; rt; s ; sti: Max Noether's Theorem. Now, Enriques- Petri's 0 ⊗2 9 Theorem. h (C; ΩC ) = 6 = 0 ⊗2 2 2 2 Hyperelliptic ) H (C; ΩC ) = hr ; rs; rt; s ; st; t i: case. 2 0 ⊗2 0 ⊗2 ; t 2 H (C; ΩC ) n H (C; ΩC (−P )) 15 / 34 The canonical map and the canonical ring of algebraic On the other hand, curves. 0 0 ⊗2 0 ⊗3 Vicente H (C; ΩC (−P )) ⊗ H (C; Ω ) ! H (C; Ω (−P )) Lorenzo C C Garc´ıa is also surjective because of the base point free pencil trick. Hence, Introduction. 0 ⊗3 3 2 2 2 2 3 2 2 Basic tools. H (C; ΩC (−P )) = hr ; r s; r t; rs ; rst; rt ; s ; s t; st i Examples. Max Noether's Now, Theorem. h0(C; Ω⊗3) = 10 9 Enriques- C = Petri's ) H0(C; Ω⊗3) = hr3; r2s; r2t; rs2; rst; Theorem. C t3 2 H0(C; Ω⊗3) n H0(C; Ω⊗3(−P )) ; Hyperelliptic C C case. rt2; s3; s2t; st2; t3i: 16 / 34 The canonical Finally, the map map and the 0 0 ⊗(n−1) 0 ⊗n canonical H (C; ΩC ) ⊗ H (C; Ω ) ! H (C; Ω ) ring of C C algebraic curves. is surjective for every n ≥ 4 because of Castelnuovo's lemma. Vicente Lorenzo Hence, Garc´ıa 0 ⊗4 4 3 3 2 2 2 2 2 3 2 2 3 4 3 2 2 3 4 Introduction. H (C; ΩC ) = hr ; r s; r t; r s ; r st; r t ; rs ; rs t; rst ; rt ; s ; s t; s t ; st ; t i: Basic tools. Examples. 0 ⊗4 Since h (C; ΩC ) = 14, Max F = A · r4 + ::: + A · t4 Noether's 1 15 Theorem. 0 ⊗4 is zero in H (C; ΩC ) for some A1;:::;A15 2 K, not all zero.
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