Beck’s theorem for plane curves

Let d ∈ Z+, K be a field of characteristic zero and A be a nonempty finite subset of K2. Denote by Cd,K the family of algebraic curves of degree d in K2 and C6d,K := ⋃d e=1 Ce,K. For any C1 ∈ Cd,K, we say that C1 is determined by A if for any C2 ∈ Cd,K such that C2 ∩A ⊇ C1 ∩A, we have that C1 = C2; we denote by Dd,K(A) the family of elements of Cd,K determined by A. Beck’s theorem establishes that if K = R and A is not collinear, then |D1,R(A)| = Θ ( |A| min C∈C1,R |A \ C| ) . In this paper we generalize Beck’s theorem showing that for all d ∈ Z+, there exists a constant c = c(d) > 0 such that if minC∈C6d,K |A \ C| > c, then |Dd,K(A)| = Θd ( |A| d ∏ e=1 ( min C∈C6e,K |A \ C| )d−e+1) . Mathematics Subject Classifications: 14N10, 52C10


Introduction
In this paper R, C, Q, Z, Z + , Z + 0 denote the set of real numbers, complex numbers, rational numbers, integers, positive integers and nonnegative integers, respectively. For any n, m ∈ Z, we write [n, m] := {k ∈ Z : n k m}. Let d, n ∈ Z + and K be a field of characteristic zero. A (plane) curve of degree d in K 2 is a subset C of K 2 which is the zero set of a polynomial in K[x, y] of degree d; we denote by C d,K the family of curves of degree d in K 2 and C d,K := d e=1 C e,K . For each nonempty finite subset A of K 2 , we say that C 1 ∈ C d,K is determined by A if for any C 2 ∈ C d,K satisfying that C 2 ∩ A ⊇ C 1 ∩ A, we have that C 1 = C 2 ; we denote by D d,K (A) the family of elements of C d,K which are determined by A. Thus, for instance, D 1,K (A) is the family of lines L in K 2 such that |L ∩ A| 2. Denote by S K the family of finite subsets of K 2 . As usual, for any maps ϕ, τ : S K → R and parameters d 1 τ (A)).
An important problem in combinatorial geometry is to know how many lines are determined by a nonempty finite subset of R 2 . P. Erdős conjectured in [8](see also [9], [10]) that if A is a nonempty finite subset of R 2 which is not collinear, then |D 1,R (A)| = Ω |A| min C∈C 1,R |A \ C| . L. M. Kelly and W. Moser proved in [15,Thm. 4.1] that if |A| = Ω(min C∈C 1,R |A \ C| 2 ), then the conjecture of Erdős holds. Later in [1], J. Beck proved the conjecture of Erdős unconditionally. Beck's theorem can be stated as follows.
Theorem 1. Let A be a finite subset of R 2 such that min C∈C 1,R |A \ C| 1. Then Beck's theorem has important applications in different areas of mathematics, and it has opened a new research field in combinatorial geometry, see for instance [1], [4], [6], [7], [13], [16]. Another important family of problems in combinatorial geometry is to bound the number of curves with a given degree that are determined by A and satisfy other conditions (for example, in the Sylvester-Gallai type results, the curves have to pass through few points of A), see for instance [2], [3], [5], [19]. Thus it seems natural and important to ask if Beck's theorem can be generalized for conics, cubics, etc. and arbitrary fields. This question is the motivation for the main result of this paper.
Theorem 2. For any d ∈ Z + , there is c 1 = c 1 (d) > 0 with the following property. Let K be a field of characteristic zero and A be a finite subset of K 2 such that min C∈C d,K |A\C| c 1 . Then A number of consequences can be obtained from Theorem 2. An immediate consequence of Theorem 2 is a lower bound of the number of curves of degree d determined by A.
Corollary 3. For any d ∈ Z + , there is c 1 = c 1 (d) > 0 with the following property. Let K be a field of characteristic zero and A be a finite subset of K 2 such that min C∈C d,K |A\C| c 1 . Then |D d,K (A)| = Ω d |A| d .
A straightforward consequence of Theorem 1 is that there is a line which contains several points of A or A determines a quadratic number of lines. Using Theorem 2, we can generalize this for curves.
Corollary 4. For any d ∈ Z + , there are c 1 = c 1 (d), c 2 = c 2 (d), c 3 = c 3 (d) > 0 with the following property. Let K be a field of characteristic zero, A be a finite subset of K 2 such that min C∈C d,K |A \ C| c 1 and e ∈ [1, d]. Then one of the following claims holds: The proof of Theorem 2 has 3 main steps that we sketch now.
• Using the Veronese map ψ d,K : , the problem of counting curves of degree d determined by A in K 2 is almost equivalent to count the number of hyper- which are generated (as flats) by ψ d,K (A). This is a consequence of Corollary 7 and Lemma 12.
• The number of hyperplanes in C d(d+3) 2 generated by ψ d,C (A) can be bounded using Lund's theorem (see Theorem 10). Lund's theorem is given in terms of essential dimension, maximal subsets with a given essential dimension, etc. Hence, to be able to conclude the proof of Theorem 2 when K = C, we need to translate the information about flats in C d(d+3) 2 provided by Lund's theorem into the information about curves in C 2 . The tools we use to do this are well known properties of the Veronese map and Bezout's theorem (see Theorem 5). The key lemma in this part of the proof is Lemma 16.
• Lund's theorem works only for K = C (and K = R) so, to conclude the proof of Theorem 2 for arbitrary fields of characteristic zero, we need a Lefschetz principle type results. In this part of the proof of Theorem 2, Lemma 18 is the result that allows us complete the proof for arbitrary fields of characteristic zero.
This paper is organized as follows. In Section 2 we state some auxiliary results. As it is explained above, we need to bound the number of hyperplanes generated by the image of A under the Veronese map, and then to translate this information into the original problem. This is done in Section 3. The conclusion of the proof of Theorem 2 is given in Section 4. Also, after we conclude the proof of Theorem 2, we discuss some facts about the constants in Theorem 2, possible generalizations, etc.

Preliminaries
In this section we state some results needed in the proof of Theorem 2. Let K be a field and n ∈ Z + . For any q(x 1 , x 2 , . . . , x n ) ∈ K[x 1 , x 2 , . . . , x n ], we denote by Z(q(x 1 , x 2 , . . . , x n )) its zero set in K n and by deg(p(x 1 , x 2 , . . . , x n )) its degree. We say that p(x, y) ∈ K[x, y] is irreducible if deg(p(x, y)) > 0 and for any factorization p(x, y) = p 1 (x, y)p 2 (x, y), we get that p i (x, y) ∈ K for some i ∈ {1, 2}. We say that Z(p(x, y)) is irreducible if p(x, y) is irreducible. We start with a weak version of Bezout's theorem.
Theorem 5. Let K be a field of characteristic zero and C 1 , C 2 ∈ C d,K be irreducible. If Proof. See [12,Cor. I.7.8].
The curves C = Z(p(x, y)) are not always uniquely determined by the polynomial p(x, y) (for instance, Z(x + y) = Z ((x + y) 2 )). However, as we will see later, the next lemma is a useful tool to know when Z(p(x, y)) = Z(q(x, y)) with p(x, y) irreducible. Lemma 6. Let K be a field and p(x, y), q(x, y) ∈ K[x, y] with p(x, y) an irreducible polynomial. If q(x, y) is not divisible by p(x, y), then Z(p(x, y)) ∩ Z(q(x, y)) is finite.
Let d ∈ Z + and K a field of characteristic zero. Write Note that for any p(x, y), q(x, y) ∈ K[x, y] such that [p(x, y)] = [q(x, y)], we have that deg(p(x, y)) = deg(q(x, y)) and Z(p(x, y)) = Z(q(x, y)); thus the map σ d,K defined below is well defined Let p(x, y) ∈ K[x, y] be such that deg(p(x, y)) > 0 and consider a factorization p(x, y) = r n i=1 p i (x, y) m i with m 1 , m 2 , . . . , m n ∈ Z + , r ∈ K, [p i (x, y)] = [p j (x, y)] for all i, j ∈ [1, n] such that i = j, and p i (x, y) irreducible for each i ∈ [1, n]. Then the irreducible curves Z(p 1 (x, y)), Z(p 2 (x, y)), . . . , Z(p n (x, y)) are known as the irreducible components of Z(p(x, y)). The irreducible components satisfy that the electronic journal of combinatorics 27(4) (2020), #P4.54 For any q(x, y) ∈ K[x, y] such that Z(p(x, y)) = Z(q(x, y)), take a factorization q(x, y) = s l i=1 q i (x, y) k i with k 1 , k 2 , . . . , k l ∈ Z + , s ∈ K, [q i (x, y)] = [q j (x, y)] for all i, j ∈ [1, l] such that i = j, and q i (x, y) irreducible for each i ∈ [1, l]. Since K is infinite, we have that Z(p 1 (x, y)), . . . Z(p n (x, y)), Z(q 1 (x, y)), . . . Z(q l (x, y)) are infinite sets. On the one hand, for each i ∈ [1, n], we have that Z(p i (x, y)) ⊆ Z(p(x, y)) = Z(q(x, y)) so Lemma 6 applied to p i (x, y) and q(x, y) implies that p i (x, y) divides q(x, y). On the other hand, for each i ∈ [1, l], we have that Z(q i (x, y)) ⊆ Z(q(x, y)) = Z(p(x, y)) so Lemma 6 applied to q i (x, y) and p(x, y) implies that q i (x, y) divides p(x, y). Hence, since K[x, y] is a unique factorization domain, we get that q(x, y) = t n i=1 p i (x, y) k i for some k 1 , k 2 , . . . , k n ∈ Z + and t ∈ K. As a consequence of these facts, the irreducible components of C do not depend on the polynomial from which C is the zero set, and we get the following corollary.
Corollary 7. Let K be field of characteristic zero, d ∈ Z + and C ∈ C d,K with pairwise distinct irreducible components Z(p 1 (x, y)), Z(p 2 (x, y)), . . . , Z(p n (x, y)). Then Since the number of solutions (m 1 , m 2 , . . . , Let K be a field and d, e ∈ Z + 0 with e d. A translation F of a vectorial subspace V of K d will be called a flat. We write dim F := dim V , and also if V is an e-dimensional subspace, we say that F is an e-flat; in particular, 1-flats are lines and d − 1-flats are hyperplanes. The family of e-flats in K d will be denoted by G e,K . For any subset S of K d , we denote by Fl(S) the smallest flat (with respect to ⊆) which contains S and we write dim S := dim Fl(S). If S = {s 1 , s 2 , . . . , s n }, we write Fl(s 1 , s 2 , . . . , s n ) := Fl(S). The family of e-flats F in K d such that there is a subset R of S satisfying that F = Fl(R) will be denoted by G e,K (S).
A fundamental tool in this paper is the Veronese map. Let d ∈ Z + and K be a field.
To avoid confusion, the ring of polynomials which corresponds to K 2 will be denoted by K[x, y] and the ring of polynomials which corresponds to K ii) The map Another important property of ψ d,K is that the image of any d + 1 elements of K 2 cannot be contained in a d − 1-flat.
Let d ∈ Z + and S be a nonempty subset of C d . The smallest e ∈ Z + 0 such that there is a collection of flats {F i } i∈I in C d satisfying that i∈I dim F i = e will be called the essential dimension of S and we will denote it by dim S. For instance, if S is the union of two skew lines in C 3 , then dim S = 3 and dim S = 2. For each e ∈ [0, d], denote by F e (S) the family of subsets R of S such that dim R e, and we write

|R|.
In other words, φ e (S) is the maximum size which can have a subset of S with essential dimension at most e. A fundamental tool in the proof of Theorem 2 is the following weak version of a theorem showed by B. Lund.
Theorem 10. For any e ∈ Z + , there is c 5 = c 5 (e) > 0 with the following property. Let d ∈ Z + be such that d e and S be a subset of C d such that |S| − φ e (S) c 5 . Then We conclude this section with a Lefschetz principle type result.
Theorem 11. Let K be a finitely generated field over Q. Then there is an injective morphism of fields ρ : K −→ C.

Curves and hyperplanes
In this section we prove some results which are needed in the proof of Theorem 2. The first result of this section shows that for any field K of characteristic zero, there is an important relation between the family of hyperplanes generated by ψ d,K (A) in K d(d+3) 2 and the family of curves of degree d in K 2 determined by A. This is done using the maps σ d,K and τ d,K defined in the previous section.
Lemma 12. Let d ∈ Z + , K be a field of characteristic zero and A be a nonempty subset of K 2 satisfying that there is no element of C d,K which contains A. Then . By contradiction, we prove that Since A is not contained in an element of C d,K by assumption, we get that A C 1 . Take and the injectivity of ψ yields that C 1 ∩ A ⊆ C ∩ A. We construct a curve C 2 as follows.
• Assume that deg(p) = d. Write C 2 := C. On the one hand, since a ∈ C 1 and a ∈ ψ −1 (H) = C, we get that C 1 = C 2 . On the other hand, In any case we constructed a (2) is true and this implies that τ ([p 1 ]) ∈ G concluding the proof of (1). Now we show that Take Since H 1 is generated as a flat by elements of ψ(A), we get that We prove by contradiction that deg( ] and then C 1 = C 2 . This shows that C 1 ∈ D, and it proves (3). The lemma is a consequence of (1) and (3).
For technical reasons, write C 0,K := ∅ for any field K.
Lemma 13. Let K be a field of characteristic zero and C be a curve in K 2 . For any and there is no element of C d,K which contains A.
Proof. Since K 2 is a surface and C a curve, we have that We assume that d > 0 from now on.
We assume that (5) is false and we will reach a contradiction. Suppose that From Remark 8.ii, we get that ψ(K 2 ) cannot be contained in a hyperplane of K d(d+3) 2 (otherwise, K 2 is contained in a curve). Thus dim ψ(K 2 ) = d(d+3)
Lemma 15. Let K be a field of characteristic zero, d, e ∈ Z + be such that e d, F be a flat in K . Then there is a curve C ∈ C e,K such that ψ d,K (C) ⊆ F and |A \ C| d ( d+2 2 )+1 .
From now on, we assume that dim F < d(d+3) 2 − 1 and set f : notice that g can be zero if ψ −1 (F ) does not contain irreducible curves. Set For each j ∈ J, we write j = (j 1 , j 2 , . . . , j f ).
Write C := Z ( g k=1 p 1,k ). Then the definition of I leads to From i) and ii), notice that for all j ∈ K, there are k, l ∈ [1, f ] such that C k,j k = C l,j l , and then Theorem 5 leads to Thus, for all j ∈ K, We get that by (17) (20) Since |A| > d ( d+2 2 )+1 , we conclude from (20) that C = ∅ (i.e. g > 0). Now we apply Lemma 14 to the polynomial g k=1 Since C ⊆ ψ −1 (F ) by (18), we have that dim ψ(C) dim F so the previous equality yields that thus deg ( g k=1 p 1,k ) e and therefore C ∈ C. This fact and (20) conclude the proof.
It looks like the upper bound |A \ C| d ( d+2 2 )+1 in Lemma 15 is not optimal. It would be an interesting problem by its own right to improve this upper bound.
Recall that if S is a subset of C d and e ∈ [1, d], then φ e (S) is the the size of a largest subset of S with essential dimension e. Lemma 16. Let d, e ∈ Z + be such that e d and A be a nonempty finite subset of C 2 . For any f ∈ d+2 Proof. Write ψ := ψ d,C , C := C e,C and φ := φ f . First we show that Take C 1 ∈ C. Then there is p(x, y) ∈ K[x, y] with deg(p(x, y)) ∈ [1, e] such that and applying Lemma 14 to the polynomial n i=1 p i , we obtain that We claim that Indeed, set F := Fl(ψ(C 1 )). Then the family (with only one flat) {F } satisfies that Since C 1 is arbitrary, (24) implies that Inasmuch as max we have that (21) is a consequence of (25). It is time to show that Fix an R ∈ F f (ψ(A)) such that φ(ψ(A)) = |R|. Since R ∈ F f (ψ(A)), we have that dim R f . Therefore we can fix a family {F i } i∈ [1,n] of flats in C For each i ∈ [1, n], let C i be the family of curves C in C 2 such that ψ(C) ⊆ F i . Relabelling if necessary, we assume that there is m ∈ [0, n] such that C i = ∅ for all i ∈ [1, m] and and fix irreducible polynomials p i,1 (x, y), p i,2 (x, y), . . . , p i,n i (x, y) ∈ C[x, y] such that Z(p i,1 ), Z(p i,2 ), . . . , Z(p i,n i ) are the pairwise distinct irreducible components of C i . We claim that for all i ∈ [1, m], If (30) is false, then the set ψ −1 (R ∩ F i ) \ C i and the flat F i satisfy the assumptions of Lemma 15 so there is a curve but this is impossible by the way we chose C i . Now we show by contradiction that for all i ∈ [m + 1, n], Indeed, if (31) does not hold, then the set ψ −1 (R ∩ F i ) and the flat F i satisfy the assumptions of Lemma 15. Therefore there is a curve The conclusion of the proof of (26) is divided into two cases.
• Suppose that m = 0. Fix any D ∈ C. Then Thus min C∈C |A \ C| |A \ D| and this completes the proof of (26) in this case.
Since the map x → d+2 and then D ∈ C.
Also note that and this completes the proof of (26).
As we explained in the introduction, Lund's theorem (i.e. Theorem 10) is not proven for arbitrary fields of characteristic zero. Therefore we need to reduce Theorem 2 to the complex case and this is what we will do in the last part of this section. Before we state and proof Lemma 18, we need some notation and observations. Let d ∈ Z + and K, L, M be fields such that M ⊆ K. For any injective morphism of fields ρ : L −→ K, abusing of notation, we denote by ρ : L d −→ K d the map (a 1 , a 2 , . . . , a d ) → (ρ(a 1 ), ρ(a 2 ), . . . , ρ(a d )). For any subset S of K[x 1 , . . . , x d ], Z K (S) will denote the common zero set of the polynomials in K d (this to distinguish in which affine space we are taking the zero set of a family of polynomials). For any subset S of K[x 1 , . . . , x d ], we say that and Proof. Since K is a field of characteristic zero, its prime subfield is isomorphic to Q so we may assume that K is an extension of Q. Let is finite, and then Lemma 12 yields that σ −1 d,K (D d,K (A)) is finite; therefore S 2 is finite. For each e ∈ [1, d], fix a curve C e ∈ C e,K such that |A \ C e | = min C∈C e,K |A \ C|, and fix q e (x, y) ∈ K[x, y] such that C e = Z K (q e (x, y)). Let S 3 be the set of all the coefficients of the polynomials in {q e (x, y) : e ∈ [1, d]}. Write L := Q(S 1 ∪ S 2 ∪ S 3 ). We have seen that S 1 , S 2 and S 3 are finite so L is finitely generated over Q. Also notice that L ⊆ K. Since the entries of the elements of A are in Q(S 1 ) ⊆ L, we get that Since M is a subfield of C, the left-hand side inequality of (46) is true. Next we prove that min Considering that |ρ(A)| = |ρ(A) \ C| + |ρ(A) ∩ C| for any curve C, we have that (47) is equivalent to max so it is enough to prove (48). Take D ∈ C e,C such that |ρ(A) ∩ D| = max C∈C e,C |ρ(A) ∩ C|. For any point there is always a curve C ∈ C e,C passing through it, then from the maximality of |ρ(A) ∩ D|, it is clear that ρ(A) ∩ D = ∅. Fix q 1 , q 2 , . . . , q n ∈ C[x, y] such that Z(q 1 ), Z(q 2 ), . . . , Z(q n ) are the pairwise distinct irreducible components of D. From Corollary 7, any q ∈ C d [x, y] such that Z(q) = D needs to satisfy that for some m 1 , m 2 , . . . , m n ∈ Z + . Hence, since D ∈ C e,C , we conclude that and then Remark 17.ii indicates that F : (note that dim F is the dimension of F as a M-flat and dim G is the dimension of G as a C-flat ). If |B| d ( d+2 2 )+1 , then (48) is true so we assume from now on that Because B ⊆ D, notice that dim G dim ψ d,C (D). Hence From (51) which implies (48) (and hence (47)). On the one hand, (36) is a direct consequence of (40), (42) and (46). On the other hand, (37) follows from (39), (41) and (43).

Proof of Theorem 2
In this section we complete the proof of Theorem 2, and then we discuss about some details of this theorem.
From Corollary 7, we get that Since C e,K ⊆ C d,K for all e ∈ [1, d] and min C∈C d,K |A \ C| c 1 by assumption, we get that min C∈C e,K |A \ C| c 1 for all e ∈ [1, d]. Then, since c 1 1, we get from (54) that for all e ∈ [1, d], satisfying that and for all i ∈ I, dim F i 1.
Theorem 9 implies that for any g ∈ [0, d−1], we have that any g-flat in C d(d+3) 2 can contain at most g + 1 elements of R; hence, since f d − 1, (61) yields that for all i ∈ I,  , by (57) and this completes the proof.
As it can be noted in the first part of the proof of Theorem 2, the constant c 1 = c 1 (d) 0 depends on the constant c 5 = c 5 d(d+3) 2 − 1 of Theorem 10. It can be noticed in [16,Sec. 7] that the constant c 5 is not easy to compute; nonetheless, Lund proves that c 5 (d) d − O(1) and he gives a conjecture of a stronger lower bound of c 5 (d).
Theorem 2 holds for fields of characteristic zero. Many tools of the proof are true also for more general fields. Nevertheless, Theorem 11 (and therefore Lemma 18) is a fundamental tool in the proof of Theorem 2. Perhaps, using some ultralimits techniques, Theorem 2 can be extended to fields with positive characteristic. Also, maybe some ideas and results established by C. Grosu in [11] are helpful to prove Theorem 2 in Z/pZ (however, it seems that Grosu's results cannot be applied directly to achieve this goal so new ideas are required).
Another interesting problem is to generalize Theorem 1.2 to higher dimensional affine algebraic subsets.