ON THE DISTRIBUTION OF RAMIFICATION POINTS IN TRIGONAL CURVES

We study the distribution of the total and ordinary ramification points of a trigonal curve over the intersection of this curve with rational curves on a rational normal scroll. We show, among other results, that these intersections may contain all the ramification points of the trigonal curve.


Introduction.
Trigonal curves are canonically immersed in rational normal scrolls, which may be viewed as the join of two rational normal curves lying in hyperplanes of complementary dimension.The purpose of this note is to find relations and bounds for the number of total and ordinary ramification points of a trigonal curve that lie on the intersection with a rational curve on the scroll.The motivation comes from results obtained by Coppens [2] relating the number of total ramification points having one or other Weierstrass sequence in a trigonal curve.In a more geometrical approach to that question, Stöhr and Viana [4] showed that a ramification point have one or other Weierstrass sequence depending on whether or not it lies on a certain rational curve (the directrix of the scroll).We use this same approach in what follows.This simplifies the treatment of the question and adds a geometrical meaning to the results (cf.[3]).

Preliminaries.
Let C be a nonsingular trigonal curve of genus g ≥ 5 defined over an algebraically closed field k of a characteristic zero and canonically embedded in P g−1 (k).We know from [1] that such a curve lies in a rational normal scroll S mn that may be described as the join of the rational normal curves and Note that each of these sets is isomorphic to the affine plane k 2 and, in [4], the following theorem was proved.
Theorem [1,Thm. 1.1].Associating to each plane curve the Zariski closure of its image under the local parametrization k 2 S mn , defined by one obtains a bijective correspondence between the canonical curves on the scroll S mn ⊂ P m+n+1 (k) and the affine plane curves given by the irreducible equation In what follows, we always denote an equation as the above by f 0 (X, Y ) = 0 and we call C 0 the plane curve defined by it.Observe that if C is the correspondent trigonal curve in S mn , then C 0 is isomorphic to the affine curve C ∩ U 0 .Furthermore, there is a plane curve which we call C n that is isomorphic to C ∩ U n under the local parametrization of S mn defined by (a : b) (a n : A point P ∈ C is a ramification point with respect to the trigonal morphism of C if the line L of the scroll passing through P is the tangent line at P , and the ramification index e P is the intersection multiplicity of C and L at P .We say that P is a total ramification point (respectively, an ordinary ramification point) when e P = 3 (respectively, when e P = 2).Observe that maps k 2 U 0 and k 2 U n+1 (respectively, k 2 U n and k 2 U g−1 ) take the vertical line X = a of k 2 into L a (respectively, into L 1/a ), a ∈ k.Thus, we call a point of C i (i = 0,n,n+ 1,g − 1) a ramification point if it has a vertical tangent line and we classify it as a total ramification point or an ordinary ramification point according to its intersection multiplicity with the tangent line as above.Equivalently, a point P = (a, b) ∈ C i (i = 0,n,n+ 1,g − 1) is a total (respectively, an ordinary) ramification point if and only if ord P (x −a) = 3 (respectively, ord P (x −a) = 2), where x is the rational function of k(C i ) = k(C), defined by (a, b) → a, and ord P (h) is the order at P of the rational function h ∈ k(C i ).
We end this section with the following observations.Remark 2.1.It is easy to check the following facts: and also a root of c 2 (X), then (a, 0) is a singular point of C n+1 , which is absurd.Similarly, if (a, 0) ∈ k 2 is a ramification point of C 0 , then a is a simple root of c 0 (X).

The main results.
The only rational curves on S mn are the lines of the ruling and the intersection of the scroll with the hyperplanes of P m+n−1 (k).We want to find bounds for the number of ramified points that lie on the intersection of a trigonal curve on the scroll with rational curves of the latter type.When m = n, all such rational curves are linearly equivalent as divisors on S mn and after a change of variables, it is enough to consider the points on C ∩ D. When m < n though, E is a distinguished curve on the scroll (e.g., is the only curve with negative self-intersection) and then we must also consider the points on C ∩ E (in this case, E is called the directrix of the scroll).
To study the occurrence of ramification points on the intersections C ∩D and C ∩E, we use the local parametrizations and the curves C 0 ,C n ,C n+1 , and C g−1 introduced above.We see that in order to count such points, it suffices to count the number of ramification points of zero ordinate in C 0 and in C n+1 , then the number of ramification points with zero abscissa on C n , and finally to check if the origin of the plane containing C g−1 is a ramification point of that curve.Proposition 3.1.Let σ (respectively, ρ) denote the number of total (respectively, ordinary) ramification points of C lying over E and let η (respectively, ξ) denote the number of total (respectively, ordinary) ramification points of C lying over D. Then one of the following possibilities occurs: In the case m < n, the above possibilities are mutually exclusive.
Proof.Suppose that c 2 ≠ 0 and c 1 ≠ 0. If there is a total ramification point in C ∩ (D\L ∞ ), then Remark 2.1(ii) implies that, for some b ∈ k, we have From all this, we may conclude that 0 ≤ σ + ρ ≤ 2m − n + 2. The other inequalities in (i) follow similarly.
To establish the inequalities in (ii), we suppose that c 2 (X) = 0 and c 1 (X) ≠ 0. Then by Remark 2.1(iv), the roots of c 3 (X) are in a one-to-one correspondence with the ramification points of C n+1 of the form (a, 0), with a ∈ k (i.e., the roots of c 3 (X) are in a one-to-one correspondence with the ramification points of C ∩ (E\L ∞ ) ).From Remark 2.3, it follows that c 3 (X) does not have a multiple root.By Remark 2.2, since c 2 (X) = 0, we must have deg c 3 (X) = 2m − n + 2 or degc 3 (X) = 2m − n + 1, but either way we have σ + ρ = 2m − n + 2. Obviously, we have ξ = 0 and 0 ≤ η + σ ≤ n + 2. Now, we do not have the restriction 0 ≤ σ + ρ + η ≤ m + 2 and to avoid an overlapping with the conditions in (i), we take η > m Using Remarks 2.1, 2.2, and 2.3 in the same way as above, one may prove (iii) and (iv), the former by assuming c 1 (X) = 0, and c 2 ≠ 0 and the latter by assuming c 1 (X) = c 2 (X) = 0.
We will show constructively that all the possibilities in Proposition 3.1 in fact occur, and now we establish some results and notation that are needed for that.If f , h ∈ k[X, Y ], we denote by Res X (f , h) the resultant of f and h as elements of (k[Y ]) [X] and Res Y (f , h) denotes the resultant of f and h as elements of (k b j X 2n−m+2−j , and f (X) = p(X)Y 3 + q(X).Suppose that p and q do not have common roots neither multiple roots.Then the set of points P such that Res X (Res Y (f , ∂f /∂X), Res Y (f , ∂f /∂Y )) ≠ 0 contains an open set of k m+n+4 .
Proof.We have Res Y (f , ∂f /∂Y ) = 27p 3 q 2 and Res Y (f , ∂f /∂X) = p 3 (dq/dX) 3 − 3qp 2 (dq/dX) 2 (dp/dX)+3q 2 pdq/dX(dp/dX) 2 −(dp/dX) 3 q 3 .From the hypothesis on p(X) and q(X), we conclude that if f is such that Res Y (f , ∂f /∂X) 0, then there is no a ∈ k satisfying both Res Y (f , ∂f /∂Y )(X) = 0 and Res Y (f , ∂f /∂X)(X) = 0, i.e., Res X (Res Y (f , ∂f /∂X), Res Y (f , ∂f /∂Y )) ≠ 0. Calculating Res Y (f , ∂f /∂X), we get where the dots indicate the sum of powers of X lesser than 3m +3n+6, A 0 = 27a  s, u, v, a, b, c, and d to grant that C is a nonsingular curve.We begin by assuming that these polynomials do not have a common root when taken two at a time and that none of them have a multiple root.We also assume that the coefficient of X 2m−n+1 in the polynomial r sa is different from zero.We have Thus, deg Res Y (f , ∂f /∂Y ) ≤ 4m + n + 10 and we assume that the coefficients of the polynomials r , s, u, v, a, b, c, and d are such that Res Y (f , ∂f /∂Y ) is a polynomial in X of degree 4m + n + 10.Observe that if we substitute b 0 and c 0, then = 27(r sa) 3 (uvd) 2 (12)   has also degree 4m − n + 10 as a polynomial in X.Similarly, we assume that the degrees of Res Y (f , ∂f /∂X) and Res Y f b=0,c=0 ,∂(f b=0,c=0 )/∂X as polynomials in X are equal to 3m + n + 9 if n ≠ m or equal to 3m + n + 6 if n = m (as in the proof of Lemma 3.2, one may check that the coefficient of X 3m+3n+9 in Res Y (f , ∂f /∂X) is equal to (n−m) times a nonidentically null polynomial expression in the leading coefficients of r , s, u, v, a, b, c, and d that does not vanish when b = c = 0; the coefficients of X 3m+3n+8 and X 3m+3n+7 are zero if n = m and the coefficient of X 3m+3n+6 is a nonzero polynomial expression in n, m and the coefficients of r , s, u, v, a, b, c, and d that does not vanish when n = m or when b = c = 0).Thus, we have Of course, we also substitute db/dx 0 and dc/dx 0 in the expression on the left side of the equality.This equality together with Lemma 3.2 show that we may choose coefficients for r , s, u, v, a, b, c, and d such that Res X Res Y (f , ∂f /∂Y ), Res Y (f , ∂f /∂Y ) ≠ 0, which implies that C 0 , and a fortiori C ∩ U 0 do not have a singular point.Observe that C ∩ L ∞ ∩ E = φ since deg(r sa) = 2m − n + 2. So, to grant the smoothness of the points of C in C ∩ L ∞ , it suffices to grant the smoothness of the points with zero abscissa of the plane curve defined by f n (X, Y ) = 0, where f n (X, Y ) is obtained from f 0 (X, Y ) by substituting X 1/X and multiplying the resulting coefficient of Y i by X 2n−m+2−i(n−m) , i = 0, 1, 2, 3. To this end, it suffices to have Res Y (f n ,∂f n /∂Y )(0) ≠ 0 but this is just another open condition on the (leading) coefficients of r , s, u, v, a, b, c, and d, and so we may take it for granted.Finally, we should check the smoothness of the points of C in C ∩ (E\L ∞ ) or, equivalently, we must check the smoothness of the points of zero ordinate of the plane curve defined by f n+1 (X, Y ) := uvdY 3 + r uvcY 2 + r subY + r sa = 0.But this is already granted because, since r sa is a polynomial without multiple roots, the curves defined by f n+1 (X, Y ) = 0 and (∂f n+1 /∂X)(X, Y ) = 0 have no points with zero ordinate in common.This shows that C is a nonsingular curve and, by Remark 3.3, C is an irreducible (trigonal, of genus m + n + 2) nonsingular curve.Using Remark 2.1, it is easy to see that C has exactly σ (respectively, ρ) total (respectively, ordinary) ramification points lying over E and η (respectively, ξ) total (respectively, ordinary) ramification points lying over D.
To find curves C with ramification points as described in the other items of Proposition 3.1 we proceed exactly as above, except for the starting curve C 0 .If σ , ρ, ξ, and η satisfy condition (ii) in Proposition 3.1, then we begin with a plane curve defined by f (X,Y ) = r sY 3 + r ucY + ud, where r , s, u, c, and d are polynomials in X of degrees σ , ρ, η, n + 2 − σ − η, and 2n − m + 2 − η respectively.If σ , ρ, ξ, and η satisfy conditions (iii) in Proposition 3.1, then we begin with a plane curve defined by f (X,Y ) = r aY 3 +r ubY 2 +uv, where r , u, v, a, and b are polynomials in X of degrees σ , η, ξ, 2m−n+2−σ , and m+2−σ −η respectively.If σ , ρ, ξ, and η satisfy conditions (iv) in Proposition 3.1, then we begin with a plane curve defined by f (X,Y ) = r Y 3 +u, where r and u are polynomials in X of degrees 2m+n−2 and 2n−m+2 respectively.Observe that the above proposition and the Riemann-Hurwitz Theorem show that there are trigonal curves C such that all the points in C ∩ D or in C ∩ E are total ramification points and these are all the ramification points of C.
The next proposition extends a result in [2], where similar bounds were obtained but only total ramification points were considered.Proposition 3.5.Let σ (respectively, ρ) be the number of total (respectively, ordinary) ramification points on E. Let τ (respectively, ς) be the number of total (respectively, ordinary) ramification points in C. Then one of the following possibilities occurs.
be the equation of the plane curve C 0 which is isomorphic to C ∩ U 0 , as we have seen in the last section, and suppose that Res 2 .From the bounds on the degrees of c 1 ,c 2 , and c 3 , we see that this polynomial has a degree which is at most 2m + 4. From Remark 2.1(iv), we have that if (a , 0) is an ordinary (respectively, a total) ramification point of C n+1 , then a is a root (respectively, is at least a double root) of 3c 1 (X)c 3 (X)−c 2 (X) 2 .In a similar way, one may see that if there is a point of the form (0,b) in C n and it is a total ramification point, then 0 is a root of 3c . Also, if the point at the origin of k 2 is an ordinary (respectively, a total) ramification point of C g−1 , then, by Remark 2.1(v), we must have that 0 is a root (respectively, is at least a double root) of 3c 1 (X)c 3 (X) − c2 (X).Note that these two last cases are mutually exclusive, for the points in the Y -axis of the plane that contains C g−1 as well as the points in the Y -axis of the plane that contains C n are all mapped into the line L ∞ through the appropriate local parametrizations of S mn described in the previous section and of course a line in the scroll may contain at most one ramification point of C.Moreover, for one of these two last cases to occur, we must have deg 3c 1 (X)c 3 (X) − c 2 (X) 2 < 2m + 4 so that anyway, we get 0 ≤ τ + 2σ + ρ ≤ 2m + 4.
Suppose now that 3c 1 (X)c 3 (X) − c 2 (X) 2 = 0. Then by Remarks 2.1(iv) and 2.3, we see that c 3 (X) does not have a multiple root and that c 3 (X) | c 1 (X).Observe that, by Remark 2. To establish the first assertion now, we use the Riemann-Hurwitz Theorem and note that it is enough to show that there is no ordinary ramification point of C in C\E.From 3c 1 (X)c 3 (X) = c 2 (X) 2 , it is easy to verify that 12c 3 (X)(∂f 0 /∂Y )(X, Y ) = (∂ 2 f 0 /∂Y 2 ) 2 (X, Y ) and also that there is no ordinary ramification point in C 0 .Finally, we note that 3c 1 (X)c 3 (X) = c2 (X) 2 , hence, 12c 3 (X)(∂f n /∂Y )(X, Y ) = (∂ 2 f n /∂Y 2 ) 2 (X, Y ).Then, as above, we conclude that there is no ordinary ramification point in C n , which completes the proof.
a and b = b/a n−m ).Similarly, we denote by f n+1 (X, Y ) = 0 the equation of the plane curve C n+1 which is isomorphic to the affine curve C ∩ U n+1 through the local parametrization of S mn , defined by (a, b) (a 0 b : ••• : a n b : a 0 : ••• : a m ), and as above we may conclude Finally, we denote by f g−1 (X, Y ) = 0 the equation of the plane curve C g−1 which is isomorphic to the affine curve C ∩ U g−1 through the local parametrization of S mn , defined by (a, b) (a n b : ••• : a 0 b : a 0 : ••• : a m ), and one may easily verify that
From Remarks 2.2 and 2.1(iii), we have that D ∩L ∞ is a total (respectively, an ordinary) ramification point of C if and only if deg c 0