Singular curves over a finite field and with many points

Recently Fukasawa, Homma and Kim introduced and studied certain projective singular curves over $\mathbb {F}_q$ with many extremal properties. Here we extend their definition to more general non-rational curves.


Introduction
Fix a prime p and a p-power q. Recently S. Fukasawa, M. Homma and S. J. Kim introduced a family of singular rational curves defined over F q , with many singular points over F q and, conjecturally, some extremal properties. In this paper we discuss a similar type of curves, discuss their extremal properties and, in some cases, show that they are, more or less, the curves introduced in [5]. The zetafunction Z Y (t) of a singular curve Y is explicitly given in terms of the Frobenius on a " topological " invariant H 1 c (Y, Q ℓ ) ( [2], [1], p. 2, [8], [12]). Hence Z Y (t) does not detect the finer invariants of the singular points of Y (it does not distinguish between unibranch points defined over the same extension of F q ; in particular it does not distinguish between a smooth point and a cusp). Using gluing of points of the normalization with the same residue field we may define a " minimal " singular curve with prescribed normalization and prescribed zeta-function.
Let Y be a geometrically integral projective curve defined over F q . Let u : C → Y denote the normalization. Since any finite field is perfect, C and u are defined over F q . Hence for every integer n ≥ 1 we have u(C(F q n )) ⊆ Y (F q n ) and for each P ∈ Y (F q n ) the scheme u −1 (P ) is defined over F q n . Hence the finite set u −1 (P ) red is defined over F q n (but of course if ♯(u −1 (P ) red ) > 1 the points of u −1 (P ) red may only be defined over a larger extension of F q ). We are interested in properties of the set Y (F q ) knowing C. A. Weil's study of the zeta-function of smooth projective curves was extended to the case of singular curves ( [2]). We will use the very useful and self-contained treatment given by Y. Aubry and M. Perret ( [1]). There are infinitely many curves Y ′ defined over F q , with C as their normalization and with the same zeta-function (see Examples 1, 2 and Lemma 2). However, given Y , there is one natural such curve if we prescribe also the sets u −1 (P ) red as subsets of [11]). We recall that Y s is an integral projective curve with C as its normalization and that u = w s • u s , where u s : C → Y s is the normalization map. Over an algebraically closed field the one-dimensional seminormal singularities with embedding dimension n ≥ 1 are exactly the singularities formally isomorphic to the local ring at the origin of the union of the coordinate axis in A n . Even over a finite field the curve we will study is defined in this way, i.e. the curves C [q,n] , n ≥ 2, defined below are obtained in the same way, i.e. the gluing process introduced by C. Traverso ([9]) gives always a seminormal curve and if the base field is algebraically closed, then all seminormal curve singularities are obtained in this way (over an algebraically closed base field a more general construction is given in [7], p. 70). We call axial singularities the curve singularities obtained in this way. Hence by definition we say that (Y, P ) is an axial singularity with embedding dimension n if and only if over F q it is formally isomorphic at P to the germ at 0 of the union of of the n axis of A n . An axial singularity of embedding dimension n > 2 is not Gorenstein. An axial singularity of embedding dimension 2 is an ordinary double point except that over a nonalgebraically closed base field, say F q , we need to distinguish if the two branches of Y at P (or the two lines of its tangent cone) are defined over F q or not (in the latter case each of them is defined over F q 2 ). Similarly, for an axial singularity (Y, P ) of embedding dimension t ≥ 2 defined over F q , the t lines of the tangent cone C(P, Y ) are defined over F q t and their union is defined over F q . In the examples we are interested in, none of these lines will be defined over a field F q e with e < t. iIf P ∈ Y is a singular point, then we may associate a non-negative integer p a (Y, P ) (usually called the arithmetic genus of the singularity or the drop of genus the singular point P ) such that p a (Y ) = p a (C) + P ∈Sing(Y ) p a (Y, P ). When Y is an axial singularity with embedding dimension n, then p a (Y, P ) = n − 1.
We fix q, C and an integer n ≥ 2. We construct a singular curve C [q,n] with C as its normalization and ♯(C [q,n] (F q )) very large in the following way. Fix an integer t such that 2 ≤ t ≤ n. For each P ∈ C(F q t ) \ C(F q t−1 ) the orbit of the Frobenius F q has order t, say {P, . . . , F t−1 q (P )}. Let C [q,n] be the only curve obtained by gluing each of these orbits (for all possible t ≤ n) (see Remark 4). By construction C [q,n] is a seminormal curve defined over F q , each singular points of C [q,n] is defined over F q and ♯(C [q,n] (F q )) = N 1 + n i=2 (N i − N i−1 )/i, where N i := ♯(C(F q i )). Fix P ∈ C [q,n] with embedding dimension t ≥ 2. The Frobenius F q acts on the local ring O C [q,n] ,P and hence on the t branches of C [q,n] at P (i.e. the t smooth branches through 0 of the tangent cone of C [q,n] at P ). Since P is an axial singularity, the action of Frobenius is the restriction to u −1 (P ) of the action of the Frobenius F q : C(F q ) → C(F q ). Hence this action is cyclic, i.e. it has a unique orbit. Thus if O = u(P ) with P ∈ C(F q t ) \ C(F q t−1 ), then p a (C [q,n] ) = t − 1 and none of the t branches of C [q,n] at u(O) is defined over a proper subfield of F q t . See Propositions 2, 3, Question 1 and Remark 2 for the relations between P 1 [q,n] and the curves B and B n studied in [5].

The curves C [q,n] and their maximality properties
Let u : C → Y denote the normalization map. We often write of Y is the product of the zeta-function Z C (t) of C and a degree ∆ Y polynomials whose inverse roots are roots of unity ( [2], [1], Theorem 2.1 and Corollary 2.4).
Let ω i , 1 ≤ i ≤ 2g, be the inverse roots of numerator of Z C (t) and β j , 1 ≤ i ≤ ∆ Y the inverse roots of the polynomial Z Y (t)/Z C (t). For every integer n ≥ 1 we have Recall that |β j | = 1 for all j. Hence among all curves with fixed normalization C and with fixed ∆ Y the integer ♯(Y (F q )) is maximal (resp. minimal) for a curve with β j = −1 for all j (resp. β j = 1) for all j), if any such curve exists. If β j = −1 for all j, then for all odd (resp. even) positive integers n among all curves with fixed normalization C and with fixed ∆ Y the value ♯(Y (F q n )) is maximal (resp. minimal). If β j = 1 for all j, then for every n > 0 the value ♯(Y (F q n )) is minimal among all curves with fixed normalization C and with fixed ∆ Y .
has all its inverse roots equal to −1 if and only if for each P ∈ Sing(Y ) either ♯(u −1 (P )) = 1 or P ∈ Y (F q ) and u −1 (P ) is formed by two points of C(F q 2 ) (in the latter case these two points are exchanged by the Frobenius and they are in is a product of polynomials, each of them associated to a different singular point of Y . Hence it is sufficient to consider separately the contribution of each singular point of Y . Fix P ∈ Sing(Y ) and call Z P (t) the associated poly- First assume ♯(u −1 (P )) = 1. The only point, We easily get that ♯(u −1 (P )) = 1 if and only if the constant 1 is the factor of Z Y (t)/Z C (t) associated to the orbit of P . Hence from now on we assume α := ♯(u −1 (P )) ≥ 2.
If d Q > d P for some Q ∈ u −1 (P ) and either d P ≥ 2 or d Q ≥ 3, then we get that (1 − t dP )Z P (t) has a root of order > max{d P , 2} and hence Z P (t) has a root = −1.
Now assume d P ≥ 2 and d Q = d P for all Q ∈ u −1 (Q). We get Z P (t) = (1−t dP ) α . Since we assumed α ≥ 2, even in this case Z P (t) has a root = −1.
Now assume d P = 1. It remains to analyze the case d Q ∈ {1, 2} for any Q ∈ u −1 (P ). If d Q = 1 for at least one Q ∈ u −1 (P ), then Remark 1. Fix q, g, C with genus g and an integer n ≥ 2.
Now assume that g > 0 and that q is a square. If C is a minimal curve for F q , then it is a minimal curve for each ). Hence for fixed q and n the integer ♯(C [q,n] (F q )) is minimal varying C among all smooth curves of genus g if and only if C is a minimal curve. Now we assume n ≥ 3 and generalize the construction of the curve C [q,n] . Fix a positive integer s ≤ n − 2 and integers t 1 , . . . , t s such that 2 ≤ t 1 < · · · < t s ≤ n.
Let C [q;t1,...,ts] be the curve obtained from C gluing only the F q -orbits of the points of C(F q t ) \ C(F q t−1 ). We get Recall that N 1 is maximal if and only if each N t with t odd is maximal and each N i with i even is minimal ( [10], [5], Theorem 10.1). Hence if all t i are odd, then ♯(C [q;t1,...,ts] ) is maximal if and only if C is a maximal curve.
In the case g = 0 we get the following result. 2] is isomorphic over F q to the plane curve B ⊂ P 2 defined in [4] and [5].
Proof. Let u : P 1 → P 1 [q,2] denote the normalization map. The normalization map Φ : P 1 → B is unramified, because the composition of it with the inclusion B ֒→ P 2 is unramified (part (i) of [5], Theorem 2.2). By [5], part (iii) of Theorem 2.2, B is a degree q + 1 plane curve with (q 2 − q)/2 singular points and Φ(P ) = Φ(Q) with P = Q if and only if u(P ) = u(Q). Hence the universal property of the seminormalization gives the existence of a morphism ψ : . Hence ψ is an isomorphism.
Proposition 3. Fix an integer n ≥ 3. Then P 1 [q,n] is the seminormalization of the curve B n ⊂ P n , defined in [5], §6, and there is a birational morphism ψ q,n : P 1 [q,n] → B n defined over F q such that ψ n,q : P 1 [q,n] (K) → B n (K) is bijective for every field K ⊇ F q .
Proof. Let u : P 1 → P 1 [q,n] and Φ n : P 1 → B n denote the normalization maps. By [5], part (ii) of Theorem 6.4, each point P ∈ Sing(B n ) corresponds to an integer t ∈ {2, . . . , n} and an orbit of the Frobenius on P 1 (F q t )\P 1 (F q t−1 ). Hence the definition of P 1 [q,n] and the universal property of the seminormalization gives a birational morphism ψ q,n : P 1 [q,n] → B n defined over F q such that ψ n,q : P 1 [q,n] (K) → B n (K) is bijective for every field K ⊇ F q . Question 1. We guess that ψ q,n is an isomorphism.
Remark 2. Fix a prime power q and the integer n ≥ 3. Let Φ n : P 1 → B n denote the normalization map. By [5], part (i) of Theorem 6.4, Φ n is unramified (this is a necessary condition for being ψ q,n an isomorphism). The following conditions are equivalent: (i) the morphism ψ q,n is an isomorphism; (ii) p a (B n ) = p a (P 1 [q,n] ); (iii) for each P ∈ Sing(B n ), say with P = Φ n (Q) and Q ∈ P 1 (F q t ) \ P 1 (F q t−1 ), the singularity (B n , P ) has arithmetic genus t − 1; (iv) for each P ∈ Sing(B n ), say with P = Φ n (Q) and Q ∈ P 1 (F q t ) \ P 1 (F q t−1 ) the tangent cone C(P, B n ) ⊂ P n is formed by t lines through P spanning a t-dimensional linear subspace.
Part (iv) is just the definition of seminormal singularity given in [2]. Since Φ n is unramified, B n has at P t smooth branches.
Proposition 4. Let C be a smooth and geometrically irreducible projective curve defined over F q . Set 2δ := ♯(C(F q 2 )) − ♯(C(F q )). Let Y a projective curve defined over F q with C as its normalization. We have ♯(Y (F q )) ≥ ♯(C(F q ))+δ and p a (Y ) ≤ g + δ if and only if Y is isomorphic to C [q,2] over F q .
Assume ♯(Y (F q )) ≥ ♯(C(F q )) + δ and p a (Y ) ≤ g + δ. Let u : C → Y be the normalization map. The morphism u is defined over F q , i.e. over a field on which Y is defined, because any finite field is perfect. We have ♯(Sing(Y )) ≤ p a (Y )−δ and equality holds only if each singular point of Y is formally isomorphic over F q either to a node or an ordinary cusp. Set , ♯(Sing(Y )(F q )) = δ and that for each P ∈ Sing(Y )(F q ) the set u −1 (P ) is formed by two points of C(F q 2 ) \ C(F q ) exchanged by the Frobenius (Lemma 1). Since p a (Y ) = g + ♯(Sing(Y )(F q )) and ♯(u −1 (P )) ≥ 2 for each P ∈ Sing(Y )(F q ), we also get that Y is nodal. Hence Y is seminormal. The structure of the fibers of u −1 (P ), Proposition 5. Let Y be a geometrical integral projective curve defined over F q and with only seminormal singularity. Let u : C → Y be the normalization map. Let ∆ Y be the degree of the polynomial 2] ) and equality holds if and only if Y is isomorphic to C [q,2] over F q .
Proof. We have ♯(Y (F q ) ≤ ♯(C(F q )) + ∆ Y and equality holds if and only if each inverse root of Z Y (t)/Z C (t) is equal to −1. Hence we may assume 2∆ Y = ♯(C(F q 2 )− ♯(C(F q )). Since Y has only seminormal singularities, u is unramified. Since u is unramified, we have ♯(u −1 (P )) ≥ 2 for all P ∈ Sing(Y ). Hence Lemma 1 gives that the fibers of u are the fibers of the normalization map C → C [q,2] . Since Y and C [q,2] are seminormal, we get that they are isomorphic. They are isomorphic over F q , because u is defined over F q and the seminormalization is defined over F q .

Remark 3.
In the case C ∼ = P 1 Propositions 4 and 5 are partial answers to a questions raised in [5], Remark 2.5. Examples 1, 2 and Lemma 2 show that we need to add some conditions on the curve Y , not only to fix the normalization P 1 and assume ∆ Y ≤ (q 2 − q)/2. Example 1. Fix a geometrically integral projective curve A defined over F q and P ∈ A(F q ). Now we define a geometrically integral curve Y defined over F q and a morphism v : Notice that for each such pair (Y, v) we would have p a (Y ) > p a (A) and that for every integer t ≥ 1 v induces a bijection To define Y and v it is sufficient to define them in a neighborhood of P in A and the glue to it the identity map A \ {P } → A \ {P }. Fix an embedding j : A ֒→ P r , r ≥ 3, and take a projection of j(A) into P 2 from an (r −3)-dimensional linear subspace not containing j(P ), but intersecting the Zariski tangent space of j(A) at j(P ).
Example 2. Fix a geometrically integral projective curve A defined over F q and any point P ∈ A reg (F q ). Let t be the minimal integer t ≥ 1 such that P ∈ A(F q t ). We assume t ≥ 2, because the case t = 1 is covered by Example 1. Hence the orbit of P by the action of the Frobenius F q has order t (it is {P, F q (P ), . . . , F t−1 q (P )}). Let Y denote the only curve and v : A → Y the only morphism obtained in the following way. We fix a bijection of sets v : A → Y and use it to define a topology on the set Y . Now we define Y as a ringed space. On Y \ u({P, F q (P ), . . . , F t−1 q (P )}) we assume that v is an isomorphism of local ringed spaces. For each Q ∈ {P, F q (P ), . . . , F t−1 q (P )} we impose that O Y,Q is the local ring of a unibranch singular point and that v is the normalization map (it may be done using the method of Example 1 with Q instead of P and F q t instead of F q ). We need to do the construction simultaneously over all Q ∈ {P, F q (P ), . . . , F t−1 q (P )} and in such a way that the morphism is defined over F q . As in Example 1 it is sufficient to define v|U , where U is a neighborhood of {P, F q (P ), . . . , F t−1 q (P )}. There is an embedding j : A → P r , r ≥ t + 2, such that the t lines T Q (j(A), Q ∈ {P, F q (P ), . . . , F t−1 q (P )}, are linearly independent. Since j is defined over F q the Frobenius F q of P r acts on j(A) and on the tangent developable of A. Since j(P ) is defined over ). Since the t tangent lines are linearly independent, the linear space E := {O, F q (O), . . . , F t−1 q (O)} has dimension t − 1. Since E is F q -invariant, it is defined over F q . Let π : P r \ E → P r−t denote the linear projection from E. Since E is defined over F q , π is defined over F q . Hence the integral projective curve T := π(j(A) \ E ∩ j(A)) ⊂ P r−t is defined over F q . Since the t tangent lines are linearly independent and O = j(P ), we have E ∩ {j(P ), . . . , j(F t−1 q (P ))} = ∅. Hence E ∩ j(U ) = ∅ for a sufficiently small neighborhood U of {P, F q (P ), . . . , F t−1 q (P )}. Assume for the moment that π|j(A) \ j(A) ∩ E is birational onto its image. Since π|j(A) \ j(A) ∩ E is birational onto its image, it is separable. Hence only finitely many points of j(A reg ) have a tangent line intersecting E. Restricting if necessary U ⊆ A reg we may assume that for no other point Q ∈ j(U )(F q ) the Zariski tangent space T j(Q) (j(A)) intersects E. Since π|j(A) \ j(A) ∩ E is birational onto its image, it is generically injective. Hence restricting U ⊆ A reg we may assume that π|j(U ) is injective and an isomorphism outside {j(P ), j(F q (P )), . . . , j(F t−1 q (P ))}. At these points the curve T has a cusp, but perhaps not an ordinary cusp, i.e. it is a unibranch singular point. Hence to conclude the example it is sufficient to find j such that π|j(A) \ j(A) ∩ E is birational onto its image. We take as j is a linearly normal embedding of degree d > max{2p a (A) − 2, p a (A) + t}. Since d > max{2p a (A) − 2, p a (A) + t}, Riemann-Roch gives r = d − p a (A). Assume that π|j(A) \ j(A) ∩ E is not birational onto its image and call x ≥ 2 its degree. Thus deg , contradicting our assumption d > p a (A) + t.
Lemma 2. Fix an integer y > 0. Let A be a geometrically integral projective curve defined over F q . Assume A reg (F q ) = ∅ and fix P ∈ A reg (F q ). Then there are a geometrically integral projective curve Y and a morphism u : A → Y defined over F q such that u induces an isomorphism of A\{P } onto Y \u(P ) and p a (Y ) = p a (A)+y.
Proof. Let m be the maximal ideal of the local ring O A,P . By assumption O A,P /m ∼ = F q and F q · 1 ⊂ O A,P . Hence the F q -vector space O A,P is the direct sum of its subspaces F q · 1 and m. Set O Y,u(P ) := F q · 1 + m y+1 ⊂ O A,P . It is easy to check that O Y,u(P ) is a local ring with m y+1 as its maximal ideal. Since P ∈ A reg , O A,P is a DVR. Hence m/m t+1 is a F q -vector space of dimension y. We take as Y the same topological space as Y , but with O Y,u(P ) at the point u(P ) associated to P instead of O A,P . With this definition of u we have dim Fq (u * (O A )/O Y ) = y. Hence p a (Y ) = p a (A) + y.
Remark 4. Fix q, C and an integer n ≥ 2. Here we explain one way to check the existence of the curve C [q,n] . We obtain C [q,n] in finitely many steps each of them similar to the one described in Example 2. We use z steps, where z is the number of orbits of F q in C(F q n ) \ C(F q ). At each of the steps we glue together one of these orbit. We do not need any notion of gluing, except that set-theoretically in each step one of these orbits is sent to a single points and for all other points the map is an isomorphism. Fix Q ∈ C(F q n ) \ C(F q ) and assume Q ∈ C(F q t ) \ C(F q t−1 ). Hence {Q, F q (Q), . . . , F t−1 q (Q)} is the orbit of Q for the action of F q . Call A the geometrically integral curve arising in the steps at which we want to glue this orbit. Hence there is a geometrically integral projective A curve defined over F q with C as its normalization (call u : C → A) and u(Q) ∈ A reg (in the previous steps (if any) the maps where isomorphism at each point of {Q, F q (Q), . . . , F t−1 q (Q)}). Set P := u(Q). Since u is defined over F q and u is an isomorphism in a neighborhood of u −1 ({P, F q (P ), . . . , F t−1 q (P )}), we have {P, F q (P ), . . . , F t−1 q (P )} ⊂ A reg and these t points are distinct. Hence P ∈ A reg (F q t ) \ A reg (F q t−1 ). As in Example 2 we get several curves Y and morphism v : A → Y defined over F q , sending {P, F q (P ), . . . , F t−1 q (P )} to a single point, O, of Y and induces an isomorphism of A \ {P, F q (P ), . . . , F t−1 q (P )} onto Y \ {O}. Let A 1 be the seminormalization of Y in A. Then we use A 1 instead of A. After z steps we get C [q,n] . To get C [q,n] we get an existence property for the seminormalization. The result does not depend from the order of the gluing. Hence C [q,n] depends only from q, C and n. Hence the curves P 1 [q,n] depends only from q and n.