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 ∈ 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 ∈ 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 ∈ 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 ϕ : 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, ∀n > 1.

Introduction. Now, if P ∈ 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 {r, s, t} 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  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 ∈ H (C, ΩC ) \ 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  Enriques- C  Petri’s ⇒ H0(C, Ω⊗3) = hr3, r2s, r2t, rs2, rst, Theorem. C t3 ∈ H0(C, Ω⊗3) \ 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 ∈ K, not all zero. Enriques- Petri’s Theorem. Since   Hyperelliptic C[r, s, t] 0 ⊗n case. dim = 4n − 2 = h (C, ΩC ), ∀n ≥ 4, (F ) n it follows that, [r, s, t] M C ' H0(C, Ω⊗n). (F ) C n≥0

17 / 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.

18 / 34 The canonical map and the canonical ring of Theorem (Max Noether) algebraic curves. If C is a non-hyperelliptic curve of genus g ≥ 4, then the canonical map Vicente M Lorenzo ϕ : S∗H0(C, Ω ) → H0(C, Ωn ) Garc´ıa C C n≥0 Introduction.

Basic tools. is surjective.

Examples. There exist P3,...,Pg ∈ C such that if we set D = P3 + ··· + Pg then, Max Noether’s • ΩC (−D) is generated by global sections. Theorem. 0 • h (C, ΩC (−D)) = 2. Enriques- 0 Petri’s We can choose P1,P2 ∈ C and a basis {ω1, . . . , ωg} of H (C, ΩC ) such that, Theorem.

Hyperelliptic  ωj (Pj ) 6= 0, case. 

 ωj (Pi) = 0 if i 6= j.

0 Moreover, H (C, ΩC (−D)) = hω1, ω2i.

19 / 34 The canonical Now, for every i ≥ 2 the base point free pencil trick implies that the following map is map and the surjective, canonical ring of algebraic 0 0 ⊗(i−1) 0 ⊗i curves. Ψi : H (C, ΩC (−D)) ⊗ H (C, ΩC ) → H (C, ΩC (−D)). Vicente Lorenzo Garc´ıa We have that: i i 0 ⊗i 0 ⊗i Introduction. • ω3, . . . , ωg ∈ H (C, ΩC ) \ H (C, ΩC (−D)). Basic tools. i i • ω3, . . . , ωg are linearly independent. Examples. • h0(C, Ω⊗i) − h0(C, Ω⊗i(−D)) = g − 2. Max C C Noether’s Theorem. Hence, H0(C, Ω⊗i) = hH0(C, Ω⊗i(−D)), ωi , . . . , ωi i. Enriques- C C 3 g Petri’s Theorem. Therefore, the following map is surjective,

Hyperelliptic 0 0 ⊗(i−1) 0 ⊗i case. H (C, ΩC ) ⊗ H (C, ΩC ) → H (C, ΩC ). It follows that, ∗ 0 M 0 ⊗n S H (C, ΩC ) → H (C, ΩC ). n≥0 is surjective. 2 20 / 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.

21 / 34 The canonical map and the canonical ring of We know that, algebraic curves. 0 ⊗2 2 2 2 2 H (C, ΩC ) = hω1 , ω1ω2, ω2 , ω1ω3, . . . , ω1ωg, ω2ω3, . . . , ω2ωg, ω3 , . . . , ωg i. Vicente Lorenzo ⊗2 Garc´ıa 0 Let i, k ∈ {3, . . . , g} be distinct. Then ωiωk ∈ H (C, ΩC ) and therefore there exist λisk, µisk, bik ∈ such that: Introduction. C

Basic tools. g X 0 ⊗2 Examples. ωiωk = bikω1ω2 + (λiskω1 + µiskω2)ωs ∈ H (C, ΩC ). s=3 Max Noether’s Theorem. It follows that

Enriques- g Petri’s X ∗ 0 Theorem. fik := ωi · ωk − bikω1 · ω2 − (λiskω1 + µiskω2) · ωs ∈ S H (C, ΩC ), Hyperelliptic s=3 case. (g−2)(g−3) is in the kernel I of ϕ. Hence, the fik’s are 2 linearly independent elements in (g−2)(g−3) the 2 -dimensional vector space I2. So, I2 is generated by the fik’s.

22 / 34 The canonical On the other hand, map and the canonical 2 2 2 2 3 2 2 3 ring of W := hω1 ω3, . . . , ω1 ωg, ω1ω2ω3, . . . , ω1ω2ωg, ω2 ω3, . . . , ω2 ωg, ω1 , ω1 ω2, ω1ω2 , ω2 i algebraic curves. 0 ⊗3 Vicente is a (3g − 2)-dimensional subspace of the (3g − 1)-dimensional space H (C, ΩC (−2D)). Lorenzo Garc´ıa 0 ⊗3 Let us take η ∈ H (C, ΩC (−2D)) \ W , so that: Introduction. 0 ⊗3 Basic tools. H (C, ΩC (−2D)) = hW, ηi. Examples. 0 For each i ∈ {3, . . . , g} we can find αi ∈ H (C, ΩC (−D)) such that Max Noether’s 2 0 ⊗3 Theorem. αiωi ∈ H (C, ΩC (−2D)) \ W. Enriques- Petri’s Therefore, there exists θi ∈ W such that Theorem. 2 Hyperelliptic αiωi = η + θi. case. It follows that given distinct k, l ∈ {3, . . . , g},

∗ 0 Gkl := αkωk · ωk − αlωl · ωl + θl − θk ∈ S H (C, ΩC ),

is in the kernel I of ϕ.It turns out that I3 is generated by the ωl · fik’s and the Gkl’s.

23 / 34 The canonical map and the canonical ring of Now we are going to see that In can be reduced to the fij ’s and the Gkl’s for every algebraic curves. n ≥ 4.

Vicente Lorenzo Firstly, by the base point free pencil trick we have that Garc´ıa 0 0 ⊗(n−1) 0 ⊗n Introduction. Θn : H (C, ΩC (−D)) ⊗ H (C, ΩC ((2 − n)D)) → H (C, ΩC ((1 − n)D)) Basic tools. is surjective for every n ≥ 4. Hence, we can prove by induction on n that, Examples.

Max 0 ⊗n l m s t h k Noether’s H (C, ΩC ((1 − n)D)) = hω1ω2 , ω1ω2ωi, ω1 ω2 η : Theorem. i ∈ {3, . . . , g}, l + m = n, s + t = n − 1, h + k = n − 3i. Enriques- Petri’s Theorem. 0 Now, for each i ∈ {1, . . . , g} let us choose βi ∈ H (C, ΩC (−D)) such that {αi, βi} Hyperelliptic 0 generates H (C, ΩC (−D)). In particular, ordPi (βi) = 1. For each j ∈ {2, . . . , g} we case. denote, n−j j n−j j Bj = {β3 ω3, . . . , βg ωg}.

24 / 34 The canonical map and the canonical ring of algebraic Given j ∈ {2, . . . , n}, we have that: curves. 0 ⊗n 0 ⊗n Vicente •B j ⊆ H (C, ΩC ((j − n)D)) \ H (C, ΩC ((j − n − 1)D)). Lorenzo Garc´ıa •B j is a set with g − 2 linearly independent elements. • h0(C, Ω⊗n((j − n)D)) − h0(C, Ω⊗n((j − n − 1)D)) = g − 2. Introduction. C C

Basic tools. Hence, for every j ∈ {2, . . . , n},

Examples. 0 ⊗n 0 ⊗n H (C, ΩC ((j − n)D)) = hH (C, ΩC ((j − n − 1)D)),Bj i. Max Noether’s Theorem. In particular, 0 ⊗n 0 ⊗n Enriques- H (C, ΩC ) = hH (C, ΩC ((1 − n)D)), B2,..., Bgi. Petri’s Theorem. This explicit basis allows us to eliminate generators of an arbitrary element of In to write Hyperelliptic it in terms of the fij ’s and the Gkl’s. case.

We conclude that I is generated by the fij ’s and the Gkl’s.

25 / 34 The canonical map and the canonical ring of algebraic curves. Theorem (Max Noether-Enriques-Petri) Vicente Lorenzo Garc´ıa Let C be a non-hyperelliptic curve of genus g ≥ 4. Then:

Introduction. (1) The following map is surjective,

Basic tools. ∗ 0 M 0 n ϕ : S H (C, ΩC ) → H (C, Ω ). Examples. C n≥0 Max Noether’s Theorem. (2) The kernel I of ϕ is generated by its elements of degree 2 and of degree 3. Enriques- (3) I is generated by its elements of degree 2 except in the following cases: Petri’s Theorem. (i) C is a nonsingular plane quintic. Hyperelliptic (ii) C is a trigonal curve. case.

26 / 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.

27 / 34 The canonical Let C be a hyperelliptic curve of genus g ≥ 3. map and the canonical 1 ring of The degree two morphism C → P is induced by a degree two invertible sheaf L such algebraic ⊗(g−1) curves. that ΩC 'L .We have that, Vicente  Lorenzo  n + 1 if 1 ≤ n ≤ g − 1, Garc´ıa h0(C, L⊗n) =

Introduction.  2n + 1 − g if n ≥ g.

Basic tools.

Examples. By the base point free pencil trick, the map

Max 0 0 ⊗(n−1) 0 ⊗n Noether’s Ψn : H (C, L) ⊗ H (C, L ) → H (C, L ) Theorem.

Enriques- is surjective for n 6= g + 1 and its image has codimension 1 in the case n = g + 1. Petri’s Theorem. 0 0 ⊗(g+1) It follows that we may choose x1, x2 ∈ H (C, L), y ∈ H (C, L ) such that Hyperelliptic 0 ⊗n case. H (C, L ) equals,

 hxn−j xj : 0 ≤ j ≤ ni if 1 ≤ n ≤ g,  1 2

 n−j j n−(g+1)−k k hx1 x2, x1 x2 y : 0 ≤ j ≤ n, 0 ≤ k ≤ n − (g + 1)i if n ≥ g + 1.

28 / 34 The canonical map and the canonical ring of algebraic curves. Since y2 ∈ H0(C, L⊗(2g+2)), we have that Vicente Lorenzo 2 2g+2 g+1 Garc´ıa F := y − (A1 · x1 + ··· + A3g+5 · yx2 )

Introduction. is zero for some A1,...,A3g+2 ∈ C not all zero. Basic tools. Examples. Since  [x , x , y]  Max dim C 1 2 = 2n + 1 − g = h0(C, L⊗n), ∀n ≥ g, Noether’s (F ) Theorem. n

Enriques- it follows that Petri’s C[x1, x2, y] M 0 ⊗n Theorem. ' H (C, L ). (F ) Hyperelliptic n≥0 case.

29 / 34 The canonical map and the canonical ring of algebraic On the other hand, we have that. curves.  g if n = 1, Vicente  Lorenzo 0 ⊗n 0 ⊗n(g−1) h (C, ΩC ) = h (C, L ) = Garc´ıa  (2n − 1)g − (2n − 1) if n ≥ 2. Introduction. Basic tools. Let us write, Examples. g−i i−1 0 ⊗(g−1) 0 • ωi := x1 x2 ∈ H (C, L ) = H (C, ΩC ), 1 ≤ i ≤ g. Max g−2−i i−1 0 ⊗2(g−1) 0 ⊗2 Noether’s • αi := x1 x2 y ∈ H (C, L ) = H (C, ΩC ), 1 ≤ i ≤ g − 2. Theorem.

Enriques- 0 ⊗n Petri’s Castelnuovo’s lemma and the explicit description of the generators of H (C, L ) yield Theorem. that, M 0 n Hyperelliptic C[ω1, . . . , ωg, α1, . . . , αg−2] → H (C, ΩC ) case. n≥0 is surjective.

30 / 34 The canonical map and the canonical ring of algebraic curves. Moreover, the kernel of this map contains the ideal I generated by: i Vicente •{ xi1 ··· x g · F : i + ··· + i = 2(g − 3)} ⊆ [ω , . . . , ω , α , . . . , α ] . Lorenzo 1 g 1 g C 1 g 1 g−2 4 Garc´ıa

Introduction. • The 2 × 2 minors of the matrix Basic tools.  ω ··· ω α ··· α  1 g−1 1 g−3 . Examples. ω2 ··· ωg α2 ··· αg−2 Max Noether’s Evenmore, counting dimensions degree by degree, we see that I coincides with the kernel Theorem. of this map and therefore, Enriques- Petri’s Theorem. C[ω1, . . . , ωg, α1, . . . , αg−2] M 0 n ' H (C, ΩC ). Hyperelliptic I case. n≥0

31 / 34 The canonical map and the Theorem canonical Let C be a hyperelliptic curve of genus g ≥ 3. Then: ring of algebraic 0 0 ⊗2 curves. (1) There exist a basis {ω1, . . . , ωg} of H (C, ΩC ) and α1, . . . , αg−2 ∈ H (C, ΩC ) Vicente such that the following map is surjective, Lorenzo Garc´ıa M 0 n ϕ : C[ω1, . . . , ωg, α1, . . . , αg−2] → H (C, ΩC ). Introduction. n≥0 Basic tools. kij ij Examples. (2) For each i, j, k ∈ {1, . . . , g − 2} there exist homogeneous f2 , f4 ∈ C[ω1, . . . , ωg] of degrees 2 and 4 respectively, such that the kernel I of ϕ is generated by: Max Noether’s Theorem. • The polynomials of the form Enriques-   Petri’s g−2 Theorem. ij X kij αiαj − f4 + f2 αk . Hyperelliptic k=1 case. • The 2 × 2 minors of the matrix

 ω ··· ω α ··· α  1 g−1 1 g−3 . ω2 ··· ωg α2 ··· αg−2

32 / 34 References

The canonical map and the canonical ring of algebraic curves.

Vicente Lorenzo Garc´ıa

Introduction. I E. Arbarello, M. Cornalba, P. Griffiths, and J. Harris, Geometry of Algebraic Curves, Springer-Verlag, 1984. Basic tools.

Examples. I R. Hartshorne, Algebraic Geometry, Springer-Verlag, 1977. Max Noether’s I B. Saint-Donat, On petri’s analysis of the linear system of quadrics through a Theorem. canonical curve, Mathematische Annalen, 206 (1973), pp. 157–175. Enriques- Petri’s Theorem.

Hyperelliptic case.

33 / 34 The canonical map and the canonical ring of algebraic curves.

Vicente Lorenzo Garc´ıa

Introduction. Basic tools. Thank you very much for your attention. Examples.

Max Noether’s Theorem.

Enriques- Petri’s Theorem.

Hyperelliptic case.

34 / 34