Divisors class groups of singular surfaces

We compute divisors class groups of singular surfaces. Most notably we produce an exact sequence that relates the Cartier divisors and almost Cartier divisors of a surface to the those of its normalization. This generalizes Hartshorne's theorem for the cubic ruled surface in P^3. We apply these results to limit the possible curves that can be set-theoretic complete intersection in P^3 in characteristic zero.


INTRODUCTION
On a nonsingular variety, the study of divisors and linear systems is classical. In fact the entire theory of curves and surfaces is dependent on this study of codimension one subvarieties and the linear and algebraic families in which they move.
This theory has been generalized in two directions: the Weil divisors on a normal variety, taking codimension one subvarieties as prime divisors; and the Cartier divisors on an arbitrary scheme, based on locally principal codimension one subschemes. Most of the literature both in algebraic geometry and commutative algebra up to now has been limited to these kinds of divisors.
More recently there have been good reasons to consider divisors on non-normal varieties. Jaffe [9] introduced the notion of an almost Cartier divisor, which is locally principal off a subset of codimension two. A theory of generalized divisors was proposed on curves in [14], and extended to any dimension in [15]. The latter paper gave a complete description of the generalized divisors on the ruled cubic surface in P 3 .
In this paper we extend that analysis to an arbitrary integral surface X, explaining the group APic X of linear equivalence classes of almost Cartier divisors on X in terms of the Picard group of the normalization S of X and certain local data at the singular points of X. We apply these results to give limitations on the possible curves that can be set-theoretic compete intersections in P 3 in characteristic zero In section 2 we explain our basic set-up, comparing divisors on a variety X to its normalization S. In Section 3 we prove a local isomorphism that computes the group of almost Cartier divisors at a singular point of X in terms of the Cartier divisors along the curve of singularities and its inverse image in the normalization. In Section 4 we derive some global exact sequences for the groups Pic X, APic X, and Pic S, which generalize the results of [15, §6] to arbitrary surfaces These results are particularly transparent for surfaces with ordinary singularities, meaning a double curve with a finite number of pinch points and triple points.
AMS 2010 Mathematics Subject Classification. Primary 14C20, 13A30; Secondary 14M10, 14J05. The second author was partially supported by the NSA and the NSF. 1 In section 5 we gather some results on curves that we need in our calculations on surfaces. Then in Section 6 we give a number of examples of surfaces and compute their groups of almost Picard divisors.
In Section 7 we apply these results to limit the possible degree and genus of curves in P 3 that can be set-theoretic complete intersections on surfaces with ordinary singularities in characteristic zero, extending earlier work of Jaffe and Boratynski. We illustrate these results with the determination of all set-theoretic complete intersections on a number of particular surfaces in P 3 .
Our main results assume that the ground field k is of characteristic zero, so that a) we can use the exponential sequence in comparing the additive and multiplicative structures, and b) so that the additive group of the field is a torsion-free abelian group.
The first author would like to thank the Department of Mathematics at the University of Notre Dame for hospitality during the preparation of this paper.

DIVISORS AND FINITE MORPHISMS
All the rings treated in this paper are Noetherian, essentially of finite type over a field k which is algebraically closed. In our application we will often compare divisors on integral surface X with its normalization S. But some of our preliminary results are valid more generally so we fix a set of assumptions.
Assumptions 2.1. Let π : S −→ X be a dominant finite morphism of reduced schemes. Let Γ and L be codimension one subschemes in S and X respectively such that π restricts to a morphism of Γ to L. Assume that S and X both satisfy G 1 (i.e. Gorenstein in codimension 1) and S 2 (i.e. Serre's condition S 2 ) so that the theory of generalized divisors developed in [15] can be applied. Further assume that the schemes Γ and L have no embedded associated primes, hence they satisfy S 1 . Now we recall the notion of generalized divisors from [15]. If X is a scheme satisfying G 1 and S 2 , we denote by K X the sheaf of total quotient rings of the structure sheaf O X . A generalized divisor on X is a fractional ideal I ⊂ K X , i.e. a coherent sub-O X -module of K X , that is nondegenerate, namely for each generic point η ∈ X, I η = K X,η , and such that I is a reflexive O X -module.
We say I is principal if it is generated by a single non-zero-divisor f in K X . We say I is Cartier if it is locally principal everywhere. We say I is almost Cartier if it is locally principal off subsets of codimension at least 2. We denote by CartX and by ACartX the groups of Cartier divisors and almost Cartier divisors, respectively, and dividing these by the subgroup of principal divisors we obtain the divisors class groups PicX and APicX, respectively. The divisor I is effective if it is contained in O X . In that case it defines a codimension one subscheme Y ⊂ X without embedded components. Conversely, for any such Y , its sheaf of ideals I Y is an effective divisor.
We recall some properties of these groups.

Proposition 2.2. Adopt assumptions 2.1. The following hold:
(a) There is a natural map π ⋆ : Pic X −→ Pic S (b) There is a natural map π ⋆ : APic X −→ APic S (c) There is an exact sequence where the sum is taken over all points x ∈ X of codimension at least two.
Proof. For (a) and (b) see [15, 2.18]. The map on Pic makes sense for any morphism of schemes. For APic, we need only to observe that since π is a dominant finite morphism, if Z ⊂ X has codimension two, then also π −1 (Z) ⊂ S has codimension two. The sequence in (c) is due to Jaffe for surfaces (see [15, 2.15]), but holds in any dimension (same proof).

Proposition 2.3. Adopt assumptions 2.1. Further assume that X and S are affine and S is smooth.
Then there is a natural group homomorphism ϕ : APicX −→ Cart Γ/π * Cart L Proof. Given a divisor class d ∈ APic X, choose an effective divisor D ∈ d that does not contain any irreducible component of L in its support (this is possible by Lemma 2.4 below). Now restrict the divisor D to X − L, transport it via the isomorphism π to S − Γ, and take its closure in S. Since S is smooth, this will be a Cartier divisor on all of S, which we can intersect with Γ to give a Cartier divisor on Γ.
If we choose another effective divisor D ′ representing the same class d, that also does not contain any component of L in its support, then D − D ′ is a principal divisor (f ) for some f ∈ K X . Since π gives an isomorphism of S − Γ to X − L it follows that S and X are birational, i.e. K X = K S . So the equation D ′ − D = (f ) persists on S, showing that the ambiguity of our construction is the Cartier divisor on Γ defined by the restriction of f . Note now that since (f ) = D − D ′ , we can write I D ′ = f I D where I D ′ and I D are the ideals of D ′ and D in O X , and f ∈ K X . If λ is a generic point of L, then after localizing, the ideals I D ′ ,λ and I D,λ are both the whole ring O X,λ , since D and D ′ are effective divisors not containing any component of L in their support. Therefore f is a unit in O X,λ . Thus f restricts to a non-zerodivisor in the total quotient ring K L , whose stalk at λ is isomorphic to O X,λ /I L,λ . Thus the restriction of f defines a Cartier divisor on L whose image in Γ will be the same as the restriction of f from S to Γ. Hence our map ϕ is well-defined to the quotient group Cart Γ/π * Cart L.
The following lemma is the affine analogue of [15, 2.11]. Lemma 2.4. Let X be an affine scheme satisfying G 1 and S 2 . Let d ∈ APic X be an equivalence class of almost Cartier divisors. Let Y 1 , . . . , Y r be irreducible codimension one subsets of X. Then there exists an effective divisor D ∈ d that contains none of the Y i in its support. Proposition 2.6. Adopt assumptions 2.1. Assume that the map induced by π from I L,X to π * (I Γ,X ) is an isomorphism. Then the map of sheaves of abelian groups γ : N −→ N 0 on X defined by the following diagram is an isomorphism: Proof. For every point x ∈ X, set (A, m A ) to be the local ring O X,x and B to be the semi-local ring O S,π −1 (x) . As it is sufficient to check an isomorphism of sheaves on stalks, we can restrict to the local situation where X = Spec A and S = Spec B. Let A 0 = A/I be the local ring of L and B 0 = B/J be the semi-local ring of Γ. Our hypothesis says that the homomorphism from A to B induces an isomorphism from I to J. Now we consider the diagram of abelian groups: and we need to show that the induced map c is an isomorphism.
Since A −→ A 0 and B −→ B 0 are surjective maps of (semi)-local rings, the corresponding maps on units a and b are surjective (see Lemma 5.2). Therefore the third map c is surjective.
To show c is injective, let a ∈ N go to 1 in N 0 . Because the diagram is commutative a comes from an element b ∈ B * and b(b) = c ∈ B * 0 whose image in N 0 is 1. Hence c comes from an element d ∈ A * 0 , which lifts to an e ∈ A * . Let f be the image of e in B * . Now b and f have the same image c in B * 0 . Regarding them as elements of the ring B this means that their difference is in the ideal J. But J by hypothesis is isomorphic to I, hence there is an element g ∈ I whose image gives b − f in J. Now consider the element h = g + e ∈ A. Since g ∈ I ⊂ m A , the element h is a unit in A, i.e. it is an element of A * . Furthermore its image in B * is b. Therefore the image of b in N , which is a, is equal to 1. Thus the map c is an isomorphism. Since this holds at all stalks x ∈ X, we conclude that the map γ : N −→ N 0 of sheaves is an isomorphism. Remark 2.7. In applications we will often consider a situation where X is an integral scheme and S is its normalization. Then O S is a generalized divisor on X, whose inverse I = {a ∈ O X | aO S ⊂ O X } is just the conductor of the integral extension. If we define L by this ideal, and Γ as π * (L), then the map induced by π from I L,X to π * (I Γ,X ) is an isomorphism and the hypothesis on the ideal sheaves is satisfied. Conversely, if the map from I L,X to π * (I Γ,X ) is an isomorphism, then Proposition 2.8. If π : S → X is a finite morphism of schemes, then the natural map is an isomorphism.
Proof. First we will show that the first higher direct image sheaf R 1 π * (O * S ) is zero. This sheaf is the sheaf associated to the presheaf which to each open subset V in X associates the group [11,III,8.1]. Hence the stalk of this sheaf at a point x ∈ X is the direct limit An element in this direct limit is represented by a pair (V, α) where V is an open set of X containing x and α ∈ H 1 (π −1 (V ), O * S | π −1 V ). This group is just Pic(π −1 V ), so the element α corresponds to an invertible sheaf L on π −1 V . We may assume that V is affine, since affine open sets form a basis for the topology. Therefore, since π is finite, the open subset π −1 V of S is also affine, and hence L is generated by global sections. Let z 1 , . . . , z r ∈ π −1 V be the finite set of points in π −1 (x). We can find a section s ∈ H 0 (π −1 V, L) that does not vanish at any of the z 1 , . . . , z r . So the zero set of s is a divisor D whose support does not contain any of the z i . Since π is finite, it is a proper morphism, so π(D) is closed in V and does not contain x. Let V ′ = V − π(D). Then L| π −1 (V ′ ) is free, and since π −1 (V ′ ) ⊂ π −1 (V ), the image of α in the above direct limit is zero. Hence R 1 π * (O * S ) = 0. Now the statement of the lemma follows from the exact sequence of terms of low degree of the Leray spectral sequence [11,III,Ex. 4.5], this means that we can also compute Pic S as H 1 (X, π * O * S ).

A LOCAL ISOMORPHISM FOR APic
In this section we prove a fundamental local isomorphism that allows us to compute the APic group of a surface locally in terms of Cartier divisors on the curves L and Γ. We first observe that if A is a local ring of dimension two satisfying G 1 and S 2 with spectrum X and punctured spectrum X ′ then APic X = Pic X ′ . Indeed APic X = APic X ′ (see [15, 1.12]), and X ′ has no points of codimension two, so APic X ′ = Pic X ′ . Theorem 3.1. Adopt assumptions 2.1. Further assume that X is the spectrum of a two dimensional local ring, S is smooth, and the map induced by π from I L,X to π * (I Γ,X ) is an isomorphism. Then the map ϕ : APicX −→ Cart Γ/π * Cart L in Proposition 2.3 is an isomorphism.
Proof. Let x be the closed point of X. Set X ′ = X − {x} and S ′ = S − {π −1 (x)}. As we noted above we can calculate APic X as Pic X ′ which is also H 1 (X ′ , O * X ′ ). We consider sheaves of abelian groups on X and similarly with primes Computing cohomology on X along (2) we obtain the exact sequence: where H 1 (X, O * X ) = Pic X = 0 since X is a local affine scheme. Now computing cohomology on X ′ along (3) we obtain the exact sequence . Now the latter is Pic S ′ , which in turn is equal to APic S. But APic S = Pic S because S is smooth and finally Pic S = 0 because S is a semi-local affine scheme.
Since X and S both satisfy S 2 any section of O X or O S over X ′ or S ′ extends to all of X or S.
. This allows us to combine the above two sequences of cohomology into one: By Proposition 2.6 applied to both maps S −→ X and S ′ −→ X ′ , we obtain H 0 (X, N ) = H 0 (X, N 0 ) and H 0 (X ′ , N ′ ) = H 0 (X ′ , N ′ 0 ). Thus we turn (4) into the following short exact sequence Using this exact sequence we can derive the following diagram: The first two rows in the diagram are obtained applying cohomology to the short exact sequences as a scheme is a disjoint union of generic points. The vertical columns arise from the fact that L and Γ are (semi)-local curves, so that when we remove the closed points we obtain the local rings of the generic points, namely the total quotient rings of O L and O Γ , and the Cartier divisors are nothing else than the quotients of the units in the total quotient rings divided by the units of the (semi)-local rings, i.e. Cart L = K * L /O * L and Cart Γ = K * Γ /O * Γ . Now the Snake Lemma yields the diagram. The last row of the above diagram implies the desired statement, namely APic X ∼ = Cart Γ/π * Cart L.

Remark 3.2.
If S is not smooth, we can replace APic X with the group G defined in Remark 2.5, in which case the proof of Theorem 3.1 shows that ϕ : G −→ Cart Γ/π * Cart L is an isomorphism, because the cokernel of H 0 (N 0 ) −→ H 0 (N ′ 0 ) is just the kernel of APic X → APic S, and in our local case, for the image an element of APic X to vanish in APic S is the same as saying it is locally free, hence it is in Pic S, which is zero because S is a semi local ring.

Proposition 3.3. If A is a local ring of dimension one satisfying S 1 and A is its completion, then
Cart A = Cart A Proof. This follows for instance from the proof of [15, 2.14] where it is shown that a → a gives a one-to-one correspondence between ideals of finite colength of A and A, under which principal ideals corresponds to principal ideals.
The following proposition shows that the local calculation of APic depends only on the analytic isomorphism class of the singularity when the normalization is smooth. Proposition 3.4. Let A be a reduced two dimensional local ring satisfying G 1 and S 2 whose normalization A is regular. Then Proof. We let S be the normalization of X = Spec A, take L to be the conductor and Γ to be π −1 (L). Then by Theorem 3.1 (cf. Remark 2.7) we can compute APic X = Cart Γ/π * Cart L. On the other hand, we have shown (Proposition 3.3) that the Cartier divisors of a one-dimensional local ring are the same as those of its completion. So applying Theorem 3.1 also to Spec A we prove the assertion.
The next example shows that we cannot drop the assumption on A being regular.

GLOBAL EXACT SEQUENCES
In this section we compare Pic and APic of any surface X to its normalization. This generalizes [15, 6.3] which dealt with the case of a ruled cubic surface. In particular our result applies to a surface with ordinary singularities whose normalization is smooth, thus providing an answer to the hope expressed in [15, 6.3.1]. Theorem 4.1. Adopt assumptions 2.1. Further assume that X is a surface either affine or projective and the map induced by π from I L,X to π * (I Γ,X ) is an isomorphism. Then there is an exact sequence: Furthermore, if S is smooth, then there is also an exact sequence Proof. For (a) we use the natural map from Proposition 2.2(a) of Pic X to Pic S and the restriction map of Pic S to Pic Γ/π * Pic L. The composition is clearly zero, since a divisor class originating on X will land in π * Pic L. To show exactness in the middle, we recall the result of Proposition 2.6 which shows that the sheaves N , N 0 in the following diagram are isomorphic: Taking cohomology on X and using Proposition 2.8, we obtain a diagram of exact sequences For (b) we first define the maps involved in the sequence. We use the natural map from Proposition 2.2(b) of APic X to APic S, which is equal to Pic S since S is smooth, together with the map ϕ of Proposition 2.3 applied locally. Note that Cart Γ/π * Cart L is simply the direct sum of all its contributions at each one of its points, since L and Γ are curves. The second map of (b) is composed of the map Pic S → Pic Γ/π * Pic L of (a) and the natural maps of Cartier divisors to Pic.
The composition of the two maps is zero, because if we start with something in APic X, then according to the construction of Proposition 2.3, its two images in Pic Γ/π * Pic L will be the same. The second map of the sequence (b) is clearly surjective.
To show exactness in the middle of (b), suppose that a class d ∈ Pic S and a divisor D in Cart Γ have the same image in Pic Γ/π * Pic L. First, we can modify D by an element of π * Cart L so that d and D will have the same image in Pic Γ. Next, by adding some effective divisors linearly equivalent to mH, where H = 0 in the affine case and H is a hyperplane section in the projective case, we can reduce to the case where d and D are effective. Consider the exact sequence of sheaves . This section defines a curve C in S in the divisor class d, not containing any component of Γ in its support, whose intersection with Γ is D. We can transport C restricted to S − Γ to X − L, and take its closure in X. This will be an element of APic X giving rise to the d and D we started with.
If X is projective, we use a hyperplane section H. Since H comes from Pic X, it is sufficient to prove the result for d + mH and D + mH. Now for m ≫ 0 the cokernel of the map H 0 (X, O S (d + mH)) → H 0 (X, O Γ (d + mH)) lands in H 1 (X, I Γ (d + mH)), which is zero by Serre's vanishing theorem. Then the proof proceeds as in the affine case.
Furthermore if S is smooth, conditions (i) and (ii) are also equivalent to In addition, without assuming S smooth, if conditions (i) and (ii) hold and the map Pic L → Pic Γ is an isomorphism then Pic X → Pic S is also an isomorphism.
Proof. From the diagram of exact sequences in the proof of Theorem 4.1(a), statement (i) is equivalent to the exactness of the sequence: , the exactness of (5) implies the exactness of Looking again at the diagram of exact sequences in the proof of Theorem 4.1(a), this implies (ii).
On the other hand, (ii) clearly implies (i). Now if S is smooth, because of the local isomorphism of Theorem 3.1, any element in the kernel of the first map of Theorem 4.1(b) is zero in all the local groups APic (Spec O X,x ), hence by Jaffe's sequence (see Proposition 2.2(c)) is already in Pic X. Thus (iii) is also equivalent to (i) and (ii). The last statement follows again from the diagram of exact sequences in the proof of Theorem 4.1(a).

Remark 4.3. If X is integral and projective in Proposition 4.2 then so is
where k is the ground field. Therefore coker(H 0 (O * X ) → π * H 0 (O * S )) = 0 and the equality of the cokernels in (ii) holds if and only if

Remark 4.4.
If S is not smooth, then as in Remarks 2.5 and 3.2 we can obtain the same results as in Theorem 4.1, with APic X replaced by G.
The next theorem shows, that at least over the complex number C, the map Pic X → Pic S is always injective. Theorem 4.5. Let X be an integral surface in P 3 over k = C. If S is the normalization of X, then the natural map Proof. From the exponential sequence [11, Appendix B, §5] we obtain an exact sequence of cohomology where X h is the associated complex analytic space of X. Now H 1 (X, O X ) = 0 since X is a complete intersection variety of dimension at least 2 [11,III,Ex. 5.5]. Furthermore H 2 (X h , Z) is a finitely generated abelian group, so we conclude that Pic X is also a finitely generated abelian group.
Next, using Grothendieck's method of comparing Pic X to the Picard group of the formal completion of P 3 along X [10, IV, §3], the proof of [10, 3.1] and the groundfield C being of characteristic zero shows that Pic X is torsion free [17,Ex. 20.7]. Thus Pic X is in fact a free finitely generated abelian group.
Taking L to be the conductor and Γ its inverse image in S, the Assumptions 2.1 are satisfied. Now, looking at the exact sequences used in the proof of Theorem 4.1, since X is integral and is equal to the kernel of the map Pic X → Pic S. If it is non zero, it must be a finitely generated free abelian group.
Since L and Γ are projective curves, the group of units in each is a direct sum of k-vector spaces and copies of k * . To see this, refer to Proposition 5.9, and note that the first sequence splits, since k * is contained in O * C . Now comparing these sequences for L and Γ we see that the cokernel of the must also be a direct sum of a k-vector space and copies of k * . But H 0 (N 0 ) = H 0 (N ) is a finitely generated free abelian group, so this cokernel must be zero.
Hence H 0 (N 0 ) is equal to the kernel of the map Pic L → Pic Γ. Since L and Γ are curves, there are degree maps on each irreducible component to copies of Z. The map Γ → L is surjective, so an element of positive degree on L remains an element of positive degree on Γ. Thus the kernel of the map Pic L → Pic Γ is just the kernel of the degree 0 part Pic 0 L → Pic 0 Γ. These are group schemes, successive extensions of abelian varieties by copies of G a and G m [20]. In particular, the kernel is also a group scheme, of finite type over k. Since H 0 (N 0 ) is a finitely generated free abelian group, as a group scheme it must have dimension zero, and hence (again using characteristic zero) must be a finite abelian group. But it is also a free abelian group, hence it is zero. Thus Pic X → Pic S is injective.

RESULTS ON CURVES
For our applications to surfaces, we need to know something about the curves L and Γ. A curve in this section will be a one dimensional scheme without embedded points, hence satisfying the condition S 1 of Serre. For any ring A, we denote by Cart A the group of Cartier divisors of Spec A. We first compute the local group Cart A at a singular point of a curve in terms of the number of local branches and an invariant δ. Then we study the Picard group of a projective curve showing the contribution of the singular points. The results of this section are essentially well-known (see the papers of Oort [19] and [20] on the construction of the generalized Jacobian). Here we gather the results on groups of divisors and divisor classes that we will need later.
In the following k * is the multiplicative group of units of the field k and k + is the whole field k as a group under addition.
Before the proof we need some Lemmas.

Lemma 5.2. Let A be a ring, a an ideal, and assume either A is a local ring or A is complete in the a-adic topology. Then the natural map of units
Proof. Let a ∈ A be an element such that a ∈ A/a is a unit. In the local case this means a ∈ m A/a , which is equivalent to saying a ∈ m A , so a is a unit.
In the complete case, there exists a b ∈ A such that ab = 1. In other words, ab = 1 + x for some x ∈ a. Now let u = 1 − x + x 2 − x 3 + . . . which exists in A since it is complete with respect to the a-dic topology. Then abu = 1, so a is a unit.
Recall that by a + we denote the ideal a as a group under addition.

Lemma 5.3. Let A be a ring and a an ideal. Assume A is complete in the a-adic topology and that
A contains the rational numbers Q. Then there is an exact sequence of abelian groups where the map α send an element a of a to exp(a) = 1 + a + 1 2 a 2 + 1 6 a 3 + . . .

Proof.
Clearly the composition βα is 1, and we know that β is surjective from the previous lemma.
If u is an element in the kernel of β that means u = 1 + y for some y ∈ a, and then is in a and gives u via the map α. This shows also that α is injective, so the sequence is exact. We need only to know that the exp map and the log map are inverses to each other and this is purely formal.
Proof of Theorem 5.1. By Proposition 3.3 we may assume that A is a complete local ring. Notice that the normalization of the completion is the completion of the normalization so A is also complete with respect to its Jacobson radical J. Let K be the total quotient ring of A. Then A ⊂ A ⊂ K, Cart A = K * /A * and Cart A = K * / A * . Therefore, we obtain the exact sequence Now A is just a product of ρ discrete valuation rings, so Cart A = Z ρ (recall that the group of Cartier divisors of a discrete valuation ring is Z). Since Z is a free abelian group, the sequence splits and we can write To analyze this latter group we will apply Lemma 5.3 to the rings A and A. Let m be the maximal ideal of A. Then we can write The first column of (6) defines an A-module M of finite length δ − ρ + 1, as is evident from considering the analogue diagram of ideals and rings without * . The first two rows of (6) are the applications of Lemma 5.3 to (A, m) and to ( A, J). The bottom row of (6) gives us an exact sequence for A * /A * . This sequence splits because A, being complete of characteristic zero, contains its residue field k. And finally, as an abelian group M is just isomorphic to (k + ) δ−ρ+1 . This gives the desired decomposition.
In the next proposition we address the non-reduced case.

Proposition 5.4. Let
A be a local ring of dimension one satisfying S 1 with residue field k of characteristic zero, let K be the total quotient ring of A, and let I be the ideal of nilpotents of A. Then there is an exact sequence Proof. We apply Lemma 5.3 to the pairs (A, I) and (K, I ⊗ K), obtaining a diagram As an abelian group (I ⊗ K)/I is a k-vector space, in particular, when we assume that char k = 0, it is a torsion-free abelian group.
(a) If A is a node, then ρ = 2 and δ = 1, so Cart A ∼ = Z 2 ⊕ k * . This result can also be deduced computationally from [15, 3.1], since any principal ideal generated by a non-zerodivisor in A = k[[x, y]]/(xy) is of the form (x r + ay s ) with a ∈ k * . Two of them multiply by adding the exponents of x and y placewise, and multiplying the coefficients of a.
(b) If A is a cusp, then ρ = 1 and δ = 1, so Cart A ∼ = Z ⊕ k + . One could also recover this result from [15, 3.7]. (c) More generally, if A is the local ring of a plane curve singularity, then ρ is the number of branches, and δ is the sum A is the local ring of a nonplanar triple point. Then we can suppose Generalizing the method of [15, 3.1], a principal ideal is generated by an element of the form x r + ay s + bz t . Under multiplication the exponents of x, y, z add, while the coefficients of y, z multiply, respectively. Hence Cart A ∼ = Z 3 ⊕ (k * ) 2 . From Theorem 5.1 we infer that δ = 2. localized at (x, y). Then A red is k[y] localized at (y). The ideal of nilpotents I is xA, which is a free module of rank 1 over A red . Thus (I ⊗ K)/I ∼ = k(y)/k[y] ∼ = y −1 k[y −1 ] is an infinite dimensional k-vector space. In the sequel we will denote this vector space by W . Now let us study H 0 (X, O * C ) and Pic C for a projective curve C.
Proposition 5.7. Suppose that C is an integral projective curve with normalization C. Then there is an exact sequence Proof. From the short exact sequence of sheaves of units taking cohomology, we obtain the long exact sequence /O * C and it has no H 1 . By Proposition 2.8, Finally, since C and C are both integral, Thus we obtain the desired short exact sequence.
Remark 5.8. Since C is a nonsingular projective curve, Pic C is an extension of Z by an abelian variety of dimension g = genus C. The dimension of Pic C is given by p a (C), the arithmetic genus of C. On the other hand, the group π * O * C /O * C is supported at a finite number of points and can be computed as in Theorem 5.1, giving where ρ i and δ i are defined at all the singular points of C. Thus reading dimensions on the short exact sequence of Proposition 5.7 we recover the well-known formula The difference of the arithmetic genus and the geometric genus is δ, which is the total number of copies of k * or k + . [11, V, 3.9.2] .
Proposition 5.9. Suppose that C is non reduced projective curve. Let I be the sheaf of nilpotent elements of C and let C red be the reduced curve. Assume that char k = 0. Then there are two exact sequences Proof. Just take cohomology of the sequence where the first map is the exponential map defined as in Lemma 5.3. Since I is a coherent sheaf on a scheme of dimension one, H 2 (I) = 0. On each connected component of C red the sections of O * C red are just k * . These sections lift to H 0 (O * C ) so the long exact sequence splits in two, and the first sequence also splits.

Example 5.10.
(a) If C is two lines in the plane P 2 meeting at a point, then C is two disjoint lines. We find that H 0 (O * C ) = k * and H 0 (O * C ) = (k * ) 2 . By Example 5.6(a) the contribution /O * C of the point is k * , and clearly Pic C = Z 2 . The exact sequence (7) then becomes hence Pic C = Z ⊕ Z (see also [15, 3.6]). (b) If C is a plane nodal cubic curve, then C is a a rational curve with Pic C = Z. The curves are integral so we can apply Proposition 5.7. The local contribution of the node (see Example 5.6(a)) is k * , so Pic C = Z ⊕ k * (see also [11,II,Ex. 6.7]). If C is a plane cuspidal cubic curve, the local contribution of the cusp is k + (see Example 5.6(b)), so as in (b) we find Pic C = Z ⊕ k + (see also [11,II,6.11.4]). (d) If C is the union of three lines in the plane meeting at a single point P , then the local contribution is (k * ) 2 ⊕ k + (see Example 5.6(c)), and C is three lines, so the exact sequence (7) is If C is the union of three lines in the plane making a triangle, then we have three local contribution of k * from the thee nodes, and using the sequence analogous to the one in (d) above, we find Pic C = Z 3 ⊕ k * . (f) Now suppose that C is the union of three lines in P 3 meeting at a point but not lying on a plane. In this case the local contribution of the point is just (k * ) 2 as we can see from Example 5.6(d). Thus as in (d) and (e) we compute Pic C = Z 3 .
Remark 5.11. Note that in all the examples above, the 'dimension' of Pic C, meaning the number of factors k * or k + plus the dimension of the abelian variety (not present in these examples) is equal to the arithmetic genus p a of the curve. The plane cubic curves have p a equal one, while the non-planar cubic curve of (f ) has p a equal zero.

EXAMPLES AND APPLICATIONS
Throughout this section, X will denote a reduced surface in P 3 and π : S → X will be its normalization. We take L to be the singular locus of X and Γ = π −1 (L). Then the hypotheses of Assumptions 2.1 are satisfied. In fact X, being a hypersurface in P 3 , is Gorenstein, and so satisfies G 1 and S 2 . We will compute L, Γ, Pic X, Pic S, and APic (Spec O X,P ) for some selected surfaces X in P 3 , to illustrate the previous theoretical material. To calculate Pic X, since X and L have each one connected component, and S and Γ have each two connected components, we have coker Now by Theorem 4.1, since Pic Γ/π * Pic L ∼ = Z and Pic S = Z ⊕ Z, we conclude that Pic X = Z. Since APic (Spec O X,x ) = Z for each point x ∈ L, the sequence of Proposition 2.2 (c) becomes where Div L is just the direct sum of a copy of Z at each point of X. Note the last map is surjective, because using a sum of lines in one of the planes H i we can get any divisor on L. Thus the sequence (8) splits and we obtain APic X ∼ = Z ⊕ Div L. ( see [15, 5.2, 5.3, 5.4] and the discussion following for more details about curves on X.) Example 6.2. Let X be the union of three planes in P 3 meeting along a line L 0 . Then S is the disjoint union of the three planes and Γ 0 is the union of one line in each of the planes. Taking L 0 and Γ 0 to be reduced, for a point P ∈ L 0 we have Cart P L 0 = Z and Cart Q i Γ 0 = Z for each of the three points Q 1 , Q 2 , Q 3 lying over P . So the quotient Cart Q Γ 0 /π * Cart P L 0 that appears in Proposition 2.3 is just Z 2 . However in this case the conductor of the normalization is not L 0 reduced! It is the scheme structure supported on the line L 0 defined by I 2 L 0 ,P 3 ( see Example 5.6(c), which shows that δ = 3 for an ordinary plane triple point).
Lifting L up to S, then Γ consists of a planar double line in each plane. Now for P ∈ L, the ideal of nilpotents I in the local ring k[x, y, z] (x,y,z) /(x, z) 2 is free of rank 2 over k[y], so using Example 5.6(e) we obtain Cart P L = Z ⊕ W 2 (where W is the k-vector space y −1 k[y −1 ]) and at each point Q i ∈ Γ above P , we find Cart Q i Γ = Z ⊕ W. The quotient Cart Q Γ/π * Cart P L is Z 2 ⊕ W . This, therefore is the group APic (Spec O X,P ), not just the Z 2 found above.
Note that if I is the sheaf of nilpotents of O L (the non-reduced structure), then I ∼ = O L 0 (−1) 2 . This sheaf has no cohomology on the line L 0 , so H 0 (O * L ) = k * and Pic L = Z. Similarly, for each component Γ i of Γ we have H 0 (O * Γ i ) = k * and Pic Γ i = Z. Now from Proposition 4.2 we conclude that Pic X −→ Pic S is injective. Looking at the first sequence of Theorem 4.1 we obtain that Pic X = Z.

Example 6.3. [Pinch points and the ruled cubic surface]
A pinch point of a surface is a singular point that is analytically isomorphic to k[x, y, z] (x,y,z) /(x 2 z − y 2 ). A typical example is the ruled cubic surface X in [11, §6]. This surface has a double line L with two pinch points. The normalization S is a nonsingular ruled cubic surface (scroll) in P 3 . The inverse of L is a conic Γ in S, and the restriction π : Γ → L is a 2-1 mapping, ramified over the two pinch points. Thus at a pinch point x ∈ L, there is just one point z ∈ Γ above x, and the mapping Cart L = Z → Cart Γ = Z is multiplication by 2. Thus APic (Spec O X,x ) = Z/2Z at the pinch point.
Since X is integral, L and Γ are both integral, and Pic L → Pic Γ is injective, Proposition 4.2 applies and we find Pic X → Pic S is injective. As Pic Γ/π * Pic L = Z/2Z, and Pic S = Z ⊕ Z, we obtain also Pic X ∼ = Z ⊕ Z.  [21], this 'superficie romana di Steiner' was discovered by Jakob Steiner in 1838 during a visit to Rome. It was first published by Kummer in 1863 in his article on quartic surfaces containing infinitely many conics, where he attributed it to Steiner. The Steiner surface X is a projection of the Veronese surface S in P 5 (the 2-uple embedding of P 2 in P 5 ), and has the equation To see this, let t, u, v be coordinate in P 2 , so that S is given by t 2 , tu, u 2 , tv, uv, v 2 in P 5 , and project by taking Then x, y, z, w satisfy the equation above. The singular locus L of X consists of the three lines x = y = 0, x = z = 0, y = z = 0 meeting at the point P = (0, 0, 0, 1) in P 3 . The curve L is a double curve for X, with two pinch points on each line. It is the conductor of the normalization π : S → X. The inverse image of L in S is a curve Γ, consisting of three conics, each meeting the other two in a point, which are the images of the three lines t = 0, u = 0, v = 0 of P 2 , forming a triangle. The three vertices of the triangle go to the triple point P , while each side of the triangle is a 2-1 covering of the corresponding line.
At the triple point P we have Cart P L = Z 3 ⊕ (k * ) 2 by Example 5.6(d). On Γ, there are three nodes Q 1 , Q 2 , Q 3 lying over P , at each of which Cart Q i Γ = Z 2 ⊕ k * by Example 5.6(a). Thus So Pic L → Pic Γ is injective and the quotient is (Z/2Z) 3 ⊕ k * .
We look at the first sequence of Theorem 4.1 Pic X −→ Pic S −→ Pic Γ/π * Pic L.
Since S is the Veronese surface in P 5 , which is the 2-uple embedding of P 2 , we have Pic S = Z, and the hyperplane section H is twice a generator. A curve C in Pic S goes to zero in Pic Γ/π * Pic L if and only if its intersection with each line of Γ is even, which means that C has even degree in P 2 , so in Pic S it is a multiple of H. Hence the kernel of Pic S → Pic Γ/π * Pic L is just Z · H. On the other hand, Proposition 4.2 applies to show that Pic X → Pic S is injective, thus Pic X = Z, generated by H. (Note that since X in P 3 is a projection of S in P 5 , the hyperplene class H on X lifts to the hyperplane class on S). Finally, we can describe APic X using Theorem 4.1(b).
Definition 6.5. We say a reduced surface X in P 3 has ordinary singularities if its singular locus L consists of a double line with transversal tangents at most points, plus a finite number of pinch points and non-planar triple points.
Remark 6.6. The significance of ordinary singularities is that one knows from the literature, at least in characteristic zero, that the generic projection X in P 3 of any nonsingular surface S in P n has only ordinary singularities [18]. In characteristic p > 0 the same applies after replacing S if necessary by a suitable d-uple embedding [22]. Conversely, if X in P 3 has ordinary singularities its normalization S is smooth (but may not have an embedding in P n of which X is the projection).

Proposition 6.7. If X is a surface in P 3 with ordinary singularities, then we can describe APic X by the sequence of Proposition 2.2(3) as
where at a pinch point of L Z 3 ⊕ k * at a triple point of L .
Proof. Indeed, the local calculation of APic is stable under passing to the completion (see Proposition 3.4) and the calculations for these three kinds of points have been done Examples 6.1, 6.2 and 6.3.

Example 6.8. [A special ruled cubic surface]
We consider the surface with equation The line L : x = y = 0 is a double line for the surface. There are no other singularities. The line L has distinct tangents everywhere except at the point P : x = y = w = 0, where it has a more complicated singularity. This surface may be regarded as a degenerate case of the ruled cubic surface considered above (see Example 6.3). It is still a projection of the cubic scroll in P 4 , but the conic Γ has become two lines.
To investigate the singularity at P , we restrict to the affine piece where z = 1, and the surface X has affine equation Adjoining u = x/y which is an integral element over the affine ring of the surface, we find that the normalization S is just k[u, w], and the map S → X is given by x = uy, y = u(w − u). Lifting the line L to S we find Γ is two lines, having equation u(w − u) = 0, and that our special point P corresponds to the origin Q : u = w = 0. The conductor of the integral extension is just L, so both L and Γ are reduced. We know that Cart Q Γ = Z 2 ⊕ k * from Example 5.6(a), and Cart P L = Z. A generator of Cart P L is given by w = 0 on L. If we lift w up to Γ it intersects each branch in one point giving (1,1) Going back to the projective surfaces X and S we know that Pic S = Z ⊕ Z, Pic Γ = Z ⊕ Z, Pic L = Z, so Pic X = Z generated by the hyperplane class H, whose image in S is (2, 1) in the notation of [15, §6]. Now by Theorem 4.1, in analogy with the ordinary ruled cubic surface above (cf. [15, 6.3]), an element of APic X can be represented by a 4-tuple (a, b, α, λ) where a, b ∈ Z, α ∈ Div L, λ ∈ k * with the condition that deg α = a − 2b.

Example 6.9. [The cone over a plane cuspidal curve]
We consider the cubic surface X given by the equation , which is the cone with vertex P = (0, 0, 0, 1) over a cuspidal curve in the plane w = 0. The normalization S is obtained by setting t = yz x . It is a cubic surface in P 4 that is the cone over a twisted cubic curve with vertex Q. The singular locus of X is the line L : x = y = 0. Its inverse image in S is the double line Γ defined by x = y = t 2 = 0. Since Γ is a planar double line, Pic L → Pic Γ is an isomorphism. Furthermore H 0 (O * L ) = k * = H 0 (O * Γ ), X and S are both integral, hence by Proposition 4.2, Pic X → Pic S is an isomorphism as well.
In place of the second sequence in Theorem 4.1, since S is not smooth, we must use the group G = ker(APic X → APic (Spec O S,Q )) (see Remark 4.4). Then Theorem 4.1 tells us that G is isomorphic to Pic S ⊕ Cart Γ/π * Cart L. Hence we have an exact sequence Here Pic S = Z, generated by three times a ruling, and for each point x ∈ L, Cart x Γ/π * Cart x L is just a group isomorphic to W = y −1 k[y −1 ] since Γ is a double line (see Example 5.6(e)).  [5] and [13] in their construction of curves in P 3 with all allowable degree and genus.) This surface can be obtained by letting S be P 2 with nine points P 0 , . . . , P 8 blown up. We take the points P i in general position, so that there is a unique cubic elliptic curve Γ passing through them. Then Pic S = Z 10 generated by a line from P 2 and the exceptional curves E 0 , . . . , E 8 . As usual, we denote by (a; b 0 , . . . , b 8 ) the divisor class al − b i E i . Thus Γ = (3; 1 9 ). Now take H = (4; 2, 1 8 ), i.e. a plane curve of degree 4 with a double point at P 0 , and passing through P 1 , . . . , P 8 . Then the complete linear system |H| maps S to a quartic surface X in P 3 , whereby the curve Γ is mapped 2 − 1 to a double line L of X with four pinch points. This is a surface X with ordinary singularities and normalization S, but it does not arise by projection from an embedding of S in some P n , because the linear system |H| is not very ample on S.
Since Γ is an elliptic curve, we have an exact sequence where Pic 0 Γ is the Jacobian variety, which in this case is just a copy of the curve itself with its group structure. If we have taken the points P i in very general position, then the restriction map Pic S → Pic Γ will be injective. Dividing by π * Pic L = Z, generated by the image of the hyperplane class H, we see that Pic X = Z, generated by H. Along the double curve L, we have APic (Spec O X,P ) = Z for a general point P , or Z/2Z at each pinch point. The surface has two disjoint double lines as its singular locus L, each having four pinch points. The inverse image Γ of L in the normalization S will be the disjoint union of two elliptic curves. We can construct this surface using an elliptic ruled surface. Following the notations of [11, V, §2], let C be an elliptic curve. Take E = O C ⊕ L 0 , where L 0 is an invertible sheaf of degree 0, not isomorphic to O C , corresponding to a divisor e on C. We take S = P(E). The map O C → E gives a section C 0 of S, so that Pic S = Z · C 0 ⊕ Pic C · f , where f is a fiber.
On the surface S we have C 2 0 = 0, C 0 · f = 1, f 2 = 0. The canonical class is K S = −2C 0 + ef . The surjection E → O C → 0 defines another section C 1 of S that does not meet C 0 . We have Now we fix a divisor class b on C of degree 2, and take H = C 0 + bf on S. Using [11,V,Ex. 2.11] we deduce that the linear system |H| has no base points. Furthermore, H 2 = 4 and H 0 (O S (H)) = 4, so the linear system |H| determines a morphism ϕ of S to P 3 , whose image we call X. One can verify (we omit the details) that H collapses the two elliptic curves C 0 and C 1 to two disjoint lines L 0 and L 1 , and otherwise is an isomorphism of S − C 0 − C 1 → X − L 0 − L 1 . So we denote by L the two disjoint lines L 0 ∪ L 1 and by Γ the two elliptic curves C 0 ∪ C 1 . The 2 − 1 coverings C 0 → L 0 and C 1 → L 1 have four ramifications points each., corresponding to four pinch points on each line. In order to find Pic X we first show that Pic S → Pic Γ is injective. Consider a divisor η = nC 0 + af in Pic S. Suppose η goes to zero in Pic Γ. On C 1 the restriction of C 0 is zero, since C 0 and C 1 do not meet. Now the image of η in Pic C 1 ⊂ Pic Γ is a, which is therefore zero. Consider the restriction of η = nC 0 to Pic C 0 . Since C 2 0 is the divisor e, the image of η is ne. If we have chosen e general, in particular a non torsion element in Pic C, then ne = 0 implies n = 0. Therefore, when we divide by π * Pic L only H vanishes. Hence Pic X = Z generated by H. As before, APic (Spec O X,P ) is Z at a general point of L, and Z/2Z at a pinch point. Example 6.12. We now consider a more complicated example. Let X be the quartic surface in P 3 defined by the polynomial f = x 4 − xyw 2 + zw 3 .
Taking partial derivatives one can verify that the singular locus L 0 of X is the line x = w = 0. Since f ∈ (x, w) 3 , the line L 0 has multiplicity 3 on X.
To find the normalization S of X we proceed as follows. This method was inspired by a computation in Macaulay 2. First consider f w 2 and let t = x 2 w . Then we find t 2 − xy + zw = 0, so t is integral over the coordinate ring of X. Next we consider z 2 f x 3 and let s = zw x . We obtain s 3 − ys 2 + xz 2 = 0, thus s is integral over X. Now the inverse image of X in the projective space P 5 with coordinates x, y, z, w, s, t is the surface S with equations We recognize this equations as the 2 × 2 minors of the 2 × 4 matrix x s w t t z x y − s .
Thus S is rational scroll of type (1, 3), namely an embedding of the rational ruled surface X 2 (in the notation of [11, V, §2]). We have Pic S = Z ⊕ Z, generated by two lines C 0 and f with intersection C 2 0 = −2, C 0 · f = 1, and f 2 = 0. The hyperplane section H is C 0 + 3f . The pullback Γ 0 of the singular locus L 0 is C 0 + 2f where C 0 = (x, w, t, y − s) and f = (x, w, s, t).
To find the conductor of the integral extension, it is enough to look at any affine piece. So let z = 1 and look at the affine ring The normalization is the affine piece of S defined by z = 1. One can eliminate variables and find its affine ring is B = k[s, y], and the map A −→ B is defined by x = s 2 (y − s), w = sx. We claim the conductor of the integral extension is just the ideal c = (x 2 , xw, w 2 ). To see this, observe that B as an A-module is generated by 1, s, s 2 . We have only to verify that the elements of c multiply s and s 2 into A. For instance, x 2 · s = xw, xw · s = w 2 , etc..
We now take the scheme L in X to be defined by the conductor so that we have an exact sequence and hence on units, using Lemma 5.3 It follows that H 0 (O * L ) = k * and Pic L = Z. Also, for any point P ∈ L, we find as in Example 5.6(e) We take Γ ⊂ S to be the pullback of L. Then Γ = 2C 0 + 4f . From the study of curves on ruled surfaces [11, V, 2.18] one knows that Γ is linearly equivalent to an irreducible nonsingular curve on S. Hence H 0 (O Γ ), which depends only on the linear equivalence class of Γ, is just k. It follows 18] so from the adjunction formula one can compute that p a (Γ) = 1. It follows that H 1 (O Γ ) = 1. Now we consider the exact sequence Since p a (Γ 0 ) = 0, we have H 0 (O Γ 0 ) = 1 and H 1 (O Γ 0 ) = 0. Now from the exact sequence of cohomology we find H 0 (I) = 0 and H 1 (I) = 1. Next, we consider the associated sequence of units Taking cohomology we obtain An analogous argument comparing Γ 0 to (Γ 0 ) red shows that Pic Γ 0 = Pic (Γ 0 ) red = Z ⊕ Z (see Example 5.10(a)). Hence we compute Pic Γ = Z ⊕ Z ⊕ k + . With the information acquired so far, we can apply Proposition 4.2 and conclude that Pic X −→ Pic S is injective. Taking for example C 0 and f as a basis for Pic S, it is clear that Pic S −→ Pic Γ is surjective. Therefore from the sequence of Theorem 4.1(a), we find that Pic X = Z and its image in S consists of those curves whose intersection numbers with the two branches of Γ, that is C 0 and f , is the same. These are the curves aC 0 + bf with b = 3a, i.e. just the multiplies of H. So Pic X = Z · H.
Finally, we will compute APic (Spec O X,P ) for a point P ∈ L. According to Theorem 3.1 this is Cart Q Γ/π * Cart P L, where Q is the point or points of Γ lying over P .
The point P 0 defined by y = 0 in L has a single point of Γ above it. All other points P ∈ L have two points lying over them. If P is general point = P 0 , then we have seen that Cart P L = Z ⊕ W 2 . The two points of Γ lying over P are on lines of multiplicity 2 and 4, respectively, so their Cart Q i Γ = Z⊕W and Z⊕W 3 respectively. Thus for a general point P ∈ L, APic (Spec O X,P ) = Z ⊕ W 2 . At the special point P 0 the situation is a bit more complicated. As before, Cart P 0 L = Z ⊕ W 2 . Using the exact sequence and Proposition 5.4 we obtain an exact sequence On the other hand, by the same method, We can also regard I as a k[y]-module. It will have rank 3, so that (I ⊗ K)/I ∼ = W 3 . Now finally, taking the quotient, we obtain

THE SEARCH FOR SET-THEORETIC COMPLETE INTERSECTIONS
We say a curve C in P 3 is a set-theoretic complete intersection (s.t.c.i. for short) if there exist surfaces X and Y such that C = X ∩ Y as sets. If the surface X containing C is already specified, we will also say C is a set-theoretic complete intersection on X.
In this section we first give some general results, building on the work of Jaffe and Boratynski [6] [7], [8], [9], [1], [2], and [3]. If C is a set-theoretic complete intersection on a surface X having ordinary singularities we show that the genus of C is bounded below as a function of its degree.
Then we examine some particular surfaces and search for all possible curves that are set-theoretic complete intersections on these surfaces.

Bounds on degree and genus.
Proposition 7.1. Let C be a curve on a surface X in P 3 that meets the singular locus in at most finitely many points. Then C is a set-theoretic complete intersection on X if and only if rC = mH in APic X, for some r , m ≥ 1, where H is a hyperplane section.
Proof. If C = X ∩ Y as sets, where Y is a surface of degree m, then the scheme X ∩ Y will be a multiple structure on the curve C. Since C is a Cartier divisor on X − Sing X, this will be rC for some r ≥ 1, so rC = mH in APic X.

Corollary 7.2.
With the hypotheses of Proposition 7.1, assume in addition that C is smooth, and that C is a set-theoretic complete intersection on X. Then at each singular point P of X lying on C, the curve C gives a non-zero torsion element in the local ring APic (Spec O X,P ).
Proof. If C is a s.t.c.i. on X, then rC ∼ mH for some r, m ≥ 1, showing that rC is a Cartier divisor. Hence the local contribution of rC at P is zero, so C is torsion in APic (Spec O X,P ). Being a smooth curve it cannot be locally Cartier at a singular point hence it is non-zero. Proposition 7.3. If C ⊂ X is a smooth curve that gives a non-zero torsion element in APic (Spec O X,P ) for some point P ∈ X, then the normalization S of X can have only one point Q lying over P .
Proof. Since rC is locally Cartier for some r ≥ 1, it will be defined locally by a single non-zero divisor f in the local ring O X,P . If S has several points Q 1 , . . . , Q s lying over P , then f will define a curve at each Q i . Thus C will have several branches and cannot be nonsingular at P . (this result is deduce by Jaffe [6, 3.3] from a more general result of Huneke.) Corollary 7.4. With the hypotheses of Corollary 7.2, assume that the surface X has only ordinary singularities. Then C can meet the singular locus of X only at pinch points.
Proof. Indeed, the normalization S of X has two points lying over a general point of L and three points lying over a triple point. (cf. Examples 6.2 and 6.3).
Proposition 7.5. With the hypotheses of Corollary 7.4, assume that char k = 0. Then already 2C = X ∩ Y for some surface Y . In this case we say that C is a self-linked curve.
Proof. We have seen that C can meet the double curve L only at pinch points. At a pinch point the local APic group is Z/2Z ( see Example 6.3), so 2C will be locally Cartier there. Thus 2C ∈ Pic X. Our hypothesis says that rC ∼ mH for some r, m ≥ 1. Thus 2C becomes a torsion element of the quotient group Pic X/Z · H. However, in case of char k = 0, this group is torsion free, by Lemma 7.6 below so we can conclude that 2C ∼ mH for some (other) m, in other words, by Proposition 7.1, already 2C is a complete intersection X ∩ Y for some Y and C is self-linked.
The statement of Lemma 7.6 is well known. We give the main steps of the proof below because of the lack of a precise reference. A proof is given in Jaffe [8, 13.2], where he assumes normality but he does not really use it. The statement is also given in [17,Ex. 20.7] as an exercise. Furthermore, a proof when X as dimension at least 3 can be found in [10,IV,3.1] (this is the proof that we mimic below). The method of proof is similar to the one employed in the proof of Theorem 4.5. Lemma 7.6. If X is a surface in P 3 over a field k of characteristic zero, then Pic X/(Z · H) is a torsion free abelian group.
Proof. In the proof of [10,IV,3.1], at each stage of the thickening, we have Pic X n+1 −→ Pic X n −→ H 2 (I n X /I n+1 X ), and the class of H comes from Pic X n+1 , so its image in the H 2 is zero. In characteristic 0, this H 2 group is torsion free, so any class in Pic X n whose multiple is in the subgroup generated by H will also go to zero in H 2 , and hence will lift to Pic X n+1 . Continuing in this way, it lifts all the way to Pic P 3 = Z, generated by H, so it is already a multiple of H.
The following result can be basically recovered from Boratynski's works ( [1] and [3]). Proposition 7.7. (Boratynski) Let C be a smooth curve in P 3 that is self-linked, so that 2C = X ∩ Y as schemes, where X and Y are surfaces of degrees m, n respectively. Then there is an effective divisor D on C such that Proof. Let C ′ be the scheme 2C. Then there is an exact sequence where L is an invertible sheaf on C. One knows from an old theorem of Ferrand that This follows also from linkage theory (see for instance [15, 4.1]), since L = I C,C ′ which is just On the other hand, L is a quotient of I/I 2 , where I = I C is the ideal sheaf of C in P 3 . Since C lies on the surface X of degree m , there is a natural map O C (−m) → I/I 2 , whose image maps to zero in L. Hence there is an effective divisor D on C such that Now from the exact sequence [11,II,8.17], taking exterior powers, we find But the above sequence with L implies that Hence O C (m − D) is a quotient of its dual N , the normal bundle of C in P 3 . Since C is smooth, N is a quotient of T P 3 |C , which in turn is a quotient of O C (1) 4 . Hence N (−1) is a sheaf generated by global sections, and therefore the same holds for O C (m − D). We conclude that the degree of this sheaf on C is non negative, i.e. deg D ≤ dm. Combining with the expression for D in the statement of Proposition 7.7 we find −4g + 4 + d(2m + n − 8) ≤ dm , which gives the first inequality.
For the second inequality, we use the fact that m + n ≥ 2 √ mn and mn = 2d since 2C = X ∩ Y .
Substituting and solving for g gives the result. Corollary 7.9. If C is a smooth curve that is a set-theoretic complete intersection on a surface X having at most ordinary singularities and ordinary nodes, and C ⊂ Sing X, and char k = 0, then the inequality of Theorem 7.8 hold, taking m = deg X and n = 2d m .

Examples and existence.
Now we will study some particular surfaces in P 3 , with the intention of finding all curves that are s.t.c.i. on them. We preserve our earlier notation: X will be a surface in P 3 , L its singular locus , S its normalization, Γ the inverse image of L, and we look for smooth curves C ⊂ X, meeting L in only finitely many points, such that C is s.t.c.i. on X. We denote by C ′ the support of the inverse of C in S, which will be a smooth curve on S isomorphic to C. We know that C can meet L only at pinch points (Corollary 7.4), so C ′ can meet Γ only at the two ramification points. In order for C to be smooth, C ′ must meet Γ transversally at these points, so the intersection multiplicity C ′ · Γ ≤ 2. Using the notation of [15, §6], the divisor class of H on S is (2, 1) and that of Γ is (1, 0). Since rC ′ = mH for some r, m, the class (a, b) of C ′ must satisfy a = 2b. But since C ′ · Γ = a ≤ 2, there is only one possibility, namely C ′ = (2, 1) = H in Pic S. Thus we see that if there is a s.t.c.i. curve C on X, it must be a twisted cubic curve, with C ′ ∼ H on S. To see if such curves C exist, we ask for a smooth curve C ′ in the linear system |H| on S that meets Γ at the two specified ramification points. Remembering that S is isomorphic to a plane P 2 blown up at one point P , and that Γ corresponds to a line ℓ not containing P , we ask for a conic C ′′ in the plane containing P and meeting ℓ at two specified points. These exist, so we conclude that there are twisted cubic curves C on X such that 2C is a complete intersection on X, and that these are the only smooth s.t.c.i. curves on X.
Example 7.13. [The special ruled cubic surface (cf. Example 6.8)] In this case the normalization S is the same cubic scroll in P 4 as in the previous example, but Γ is now two lines meeting at the point Q that lies over the special point P on L. If C is a s.t.c.i. curve on X, then by Proposition 7.3 it can meet L only at P , and since C is smooth, C ′ must meet Γ at Q without being tangent to either of the two lines of Γ. As in the previous example, the class of H in Pic S is (2, 1), the class of Γ is (1, 0), the intersection number C ′ · Γ = 2, and we find that the only possibility is C ′ ∼ H, so C will be a twisted cubic curve.
There is one well-known example of such a curve [12, Note, p. 381], namely (modulo slight change of notation) the twisted cubic curve C whose affine equation in the open set z = 1 is given parametrically by For this curve, 2C is the complete intersection of X with the surface w 2 = 4yz.
If we modify these equations by inserting a parameter λ = 0, ±i, then the curve C λ defined by λ t is another smooth twisted cubic curve lying on X and passing though the point P , and C ′ λ on S is still the linear system |H|.
To see if C λ is a s.t.c.i. on X, we must find out if C λ gives a torsion element in the local group APic (Spec O X,P ), which, according to Example 6.8, is isomorphic to Z ⊕ k * . Since C λ meets each branch of Γ just once, the Z component of C λ in APic is 0.
To find the element of k * representing C λ in APic , we recall from Example 5.6(a) that if A ∼ = k[[x, y]]/(xy), then an ideal (x r + ay s ) gives the element (r, s; a) in Cart A ∼ = Z 2 ⊕ k * . Since Γ has local equation u(w − u) = 0, it is convenient to make a change of variables v = w − u. then Γ is defined by uv = 0, and the curve C ′ λ is defined by so the local equation of C λ is u − λ 2 v, and we find that C λ gives the element µ = −λ 2 in k * . The torsion elements in k * are just the roots of unity. Note that the first curve we discussed above, which corresponds to λ = 1, gives µ = −1, which is a torsion element of order 2 in k * , confirming that 2C is a complete intersection.
For the other curves C λ in this algebraic family, we see that for a general λ, the curve C λ is not a s.t.c.i. on X, but if λ is a root of unity, then C λ will be a s.t.c.i. For any n ≥ 2, taking µ to be a primitive n th root of unity, and solving λ 2 = −µ, we find a curve C λ that is a s.t.c.i. on X of order n, but of no lower order. Note, by the way, that even though the C ′ λ are all linearly equivalent to H on S, the curves C λ 1 and C λ 2 on X are not linearly equivalent in APic X (unless λ 1 = −λ 2 , in which case the curves C λ 1 and C λ 2 are the same, replacing t by −t) because this class, in the notation of Example 6.8, is (2, 1, 0, µ). Remark 7.14. In particular the set of curves in an algebraic family on X that are s.t.c.i. need not be either open or closed.

Example 7.15. [The Steiner surface (cf. Example 6.4)]
Here the singular locus L is three lines meeting at a point, and each having two pinch points. The normalization S is isomorphic to P 2 , so Pic S ∼ = Z, generated by a line ℓ. The hyperplane section H on X corresponds to the divisor class 2ℓ on S. The curve Γ on S corresponds to three lines forming a triangle in P 2 .
A s.t.c.i. curve C in X can meet L only at pinch points (Corollary 7.4), so the curve C ′ ⊂ S meets Γ only at the ramification points. Thus C ′ · Γ ≤ 6. On the other hand, C ′ ∼ aℓ for some a ≥ 1, and ℓ · Γ = 3, so a = 1 or 2. Thus we are looking for a line or a conic in P 2 that meets the triangle Γ only in the six ramification points.
On the other hand, since these six ramification points are aligned 3 at the time, there is no smooth conic passing through all 6. Thus the four conics just mentioned are the only s.t.c.i. curves on X.
Example 7.16. [A rational quartic surface with a double line (cf. Example 6.10)] In this case Γ is an elliptic curve mapping 2-1 to the line L, so there are four ramification points. We have H · Γ = 2, so we look for curves C ′ ∼ H or C ′ ∼ 2H meeting Γ only in the ramification points. In the linear system |H| on S, every divisor meets Γ in a pair of the involution σ defining the map Γ → L. Thus it cannot meet Γ in two ramification points. The divisor 2H meets Γ in the linear system 2σ. If we compare the map Γ → L to the standard double covering of the x−axis by the curve y 2 = x(x − 1)(x − λ) [11, Chapter IV], then σ is just pairs of points that add up to 0 in the group law on the cubic curve, and the four ramification points are the points of order two in the group law. These form a subgroup isomorphic to the Klein four group Z/2Z ⊕ Z/2Z, and the sum of all four elements of this group is zero. Hence the sum of the four ramification points is in the linear system defined by 2H on Γ. Now a straightforward computation of cohomology groups shows that H 0 (O S (2H)) → H 0 (O Γ (2H)) is surjective, so there exist curves C ′ ∼ 2H meeting Γ just at the four ramification points, and it is easy to see, using Bertini's theorem, that we may take C ′ to be smooth. Thus we find smooth curves C of degree 8 and genus 7 on X for which 2C is a complete intersection, and these are the only s.t.c.i. on X.
Example 7.17. [A quartic surface with two disjoint double lines (cf. Example 6.11) ] Here the singular locus L is two lines, and Γ is two disjoint elliptic curves, each having four ramification points. Since H · Γ = 4, we look for curves C ′ ∼ H or C ′ ∼ 2H meeting Γ only in the ramification points. Any curve C ′ ∼ H maps to a singular curve in X, so we eliminate this case. Consider C ′ ∼ 2H. As in the previous example, the sum of the ramification points on Γ is in the linear system induced on Γ by 2H. Again, standard calculations of cohomology together with Bertini's theorem show that we can find smooth curves C ′ ∼ 2H meeting Γ only at the ramification points. These give smooth curves C ⊂ X of degree 8 and genus 5 for which 2C is a complete intersection, and these are the only s.t.c.i. on X.
Example 7.18. [The quartic surface of Example 6.12] We will show that there are no smooth s.t.c.i. curves on this surface. By a reasoning similar to the previous examples we are looking for a smooth curve C ′ ⊂ S meeting Γ only at the special point and not tangent to either branch of Γ. So its local equation will be y − λs+ higher terms, for some λ = 0, 1. For this computation it will be sufficient to use the reduced line L 0 and its inverse image Γ 0 defined by s 2 (y−s) = 0. We will use Proposition 2.3, and show that the image in Cart Γ 0 /π * Cart L 0 of the class of C ′ in APic (Spec O X,P 0 ) is not torsion. In Example 6.12, we found Cart Q 0 Γ 0 = Z 2 ⊕ k 8 ⊕ W . Since Cart P 0 L 0 = Z, we are looking at the group Z ⊕ k * ⊕ W . Now according to Proposition 5.4 and Example 5.6(e), we write y − λs+higher terms as y(1 − λsy −1 + higher), and find that the contribution of C ′ to W is −λ.
The group law in W is addition. It is a k-vector space, and in characteristic zero, it is torsion free. Thus C ′ gives a non-zero element, and no multiple is zero, so C cannot be a s.t.c.i.
Recall that the curves that are self-linked in P 3 are s.t.c.i. and conversely on a surface with ordinary singularities in characteristic zero any smooth s.t.c.i. curve not contained in the singular locus is self-linked. These are the only possibilities, except maybe for (d, g) = (8, 3), which seems unlikely, but we cannot yet exclude. Notes                    a) strict complete intersection b) 2C is a complete intersection on a quadratic cone c) Gallarati [4] finds this one on a cubic surface with four nodes d) Gallarati finds these on quartic surfaces with nodes. The case of g = 5 lying on a Kummar surface was known to Humbert (1883). We find both cases on quartic surfaces with ordinary (singularities Examples 7.16, 7.17).
. Remark 7.20. The curve of degree 8 and genus 5 is the first known example of a non-arithmetically Cohen-Macaulay curve in P 3 that is a set-theoretic complete intersection in characteristic zero. (In characteristic p > 0 there are many [12]).
Remark 7.21. using the same techniques, we looked for smooth s.t.c.i. curves on a surface X that is generic projection of a smooth surface S in P n , for some well-known surfaces S. We assume always that C is not contained in the singular locus of X, and that char k = 0. We summarize the results here without the computations.
(1) If S is a del Pezzo surface of degree n with 4 ≤ n ≤ 9, there are no s.t.c.i. curves on X except in the case n = 4, where there are quartic elliptic curves C with 2C a complete intersection. Of course these curves are already strict complete intersection in P 3 . (2) If S is the quintic elliptic scroll in P 4 , there are no s.t.c.i. curves on X.
(3) Suppose S is a rational scroll S e,n for any n > e ≥ 0 of degree d = 2n − e and having an exceptional curve C 0 with C 2 0 = −e. In case d = 3, we obtain the ruled cubic surface studied above (Example 7.12). For all d ≥ 4 there are no s.t.c.i. curves on X.
(4) Suppose S is the n-uple embedding of P 2 in P N , for n ≥ 2. In case n = 2 we obtain the Steiner surface discussed in Example 7.15. If n ≥ 3, there are no smooth s.t.c.i. curves on X.
Our conclusion is that smooth s..t.c.i. curves on surfaces with ordinary singularities are rather rare!