The different of a smooth plane curve #
Let g ∈ k[X, Y] be a polynomial whose affine curve g = 0 is smooth, in the ideal-theoretic
sense that g, ∂g/∂X and ∂g/∂Y generate the unit ideal, and which is monic as a polynomial in
Y over k[X]. Let K = k(t) be a rational function field and F a finite separable extension
of K generated by an element y with g(t, y) = 0, where g has degree [F : K] in Y. This
file first computes the different exponents of F / K at the places of F at which t is
regular:
d(P ∣ P ∩ K) = ord_P (∂g/∂Y (t, y)).
So such a place ramifies exactly where ∂g/∂Y vanishes on the curve.
The proof identifies the coordinate ring. Since g is monic in Y, y is integral over
k[t], and comparing degrees shows that its minimal polynomial is g(t, Y), so
k[t][y] ≅ k[X, Y] ⧸ (g). The smoothness hypothesis makes this ring a Dedekind domain
(MvPolynomial.isIntegrallyClosed_adjoin_singleton), in particular integrally closed, and its
fraction field is F. Hence k[t][y] is the integral closure of k[t] in F, generated by y,
and its different ideal is generated by ∂g/∂Y (t, y)
(TauCeti.Place.differentExponent_eq_ord_aeval_derivative_minpoly_of_isIntegralClosure).
The whole curve #
Now take K = k(x) and let g be irreducible of total degree n, with the coefficient of Yⁿ
equal to 1.
Then the projective closure of g = 0 does not pass through (0 : 1 : 0), [F : k(x)] = n, and
the places of F at which x has a pole are the points at infinity, which lie in the affine
chart with coordinates u = 1 / x and v = y / x and equation
h = MvPolynomial.inftyChart n g = uⁿ g(1 / u, v / u). The field k(u) is k(x), so if this
chart is smooth as well, the first computation for t = u gives the different exponents above
infinity. Since ∂h/∂v (u, v) = u^(n - 1) ∂g/∂Y (x, y), both charts are covered by the single
divisor identity
Diff(F / k(x)) = div (∂g/∂Y (x, y)) + (n - 1) (x)_∞.
The principal divisor has degree zero and the pole divisor of x has degree [F : k(x)] = n, so
deg Diff(F / k(x)) = n (n - 1). The genus (n - 1)(n - 2) / 2 of a smooth plane curve of degree
n follows from this by the Hurwitz genus formula, in
TauCeti.FieldTheory.FunctionField.PlaneCurve.Genus.
Main results #
TauCeti.Place.mem_integers_of_aeval_eq_zero: a rootbofφ(a, Y), forφmonic inY, is regular whereverais.TauCeti.Place.differentExponent_eq_ord_aeval_pderiv_one: at a place at whichtis regular, the different exponent ofF / k(t)is the order of∂g/∂Y (t, y).TauCeti.minpoly_ratFunc_eq_map_of_aeval_eq_zeroandTauCeti.finrank_ratFunc_eq_natDegree_of_aeval_eq_zero: for irreducibleg, the minimal polynomial ofyoverk(x)isg(x, Y), so[F : k(x)]is the degree ofginY.TauCeti.finrank_ratFunc_eq_totalDegree_of_aeval_eq_zero: if moreover the coefficient ofYⁿingis1, wherenis the total degree ofg, then[F : k(x)] = n.TauCeti.Place.differentExponent_eq_ord_aeval_pderiv_one_sub_mul_ord: at a place at whichxhas no zero,d(P ∣ P ∩ k(x)) = ord_P (∂g/∂Y (x, y)) - (n - 1) ord_P x.TauCeti.Divisor.different_eq_principal_add_zsmul_poles:Diff(F / k(x)) = div (∂g/∂Y (x, y)) + (n - 1) (x)_∞.TauCeti.Divisor.degree_different_eq_totalDegree_mul:deg Diff(F / k(x)) = n (n - 1).
References #
- H. Stichtenoth, Algebraic Function Fields and Codes, 2nd ed., GTM 254, Springer, 2009, Theorem 3.5.10.
- W. Fulton, Algebraic Curves, Section 3.2, for the local rings of a smooth plane curve.
A root of a polynomial monic in Y is regular wherever its first coordinate is: if
φ ∈ k[X, Y] is monic as a polynomial in Y over k[X], φ(a, b) = 0, and a is regular at the
place P, then so is b, which is integral over k[a].
The different of a smooth affine plane curve over a rational function field: let
K = k(t) with t transcendental over k, let g ∈ k[X, Y] be monic in Y over k[X] of
degree [F : K], with g, ∂g/∂X and ∂g/∂Y generating the unit ideal, and let F / K be
generated by y with g(t, y) = 0. At every place P of F at which t is regular, the
different exponent of F / K is ord_P (∂g/∂Y (t, y)).
The whole plane curve over k(x) #
The minimal polynomial of a point of a plane curve over k(x): an irreducible
g ∈ k[X, Y], monic in Y over k[X], with g(x, y) = 0, read as a polynomial in Y over
k(x), is the minimal polynomial of y.
The degree of a plane curve over k(x): if F / k(x) is generated by y with
g(x, y) = 0, where g ∈ k[X, Y] is irreducible and monic in Y over k[X], then [F : k(x)]
is the degree of g in Y.
The degree of a plane curve of degree n over k(x): if F / k(x) is generated by y
with g(x, y) = 0, where g ∈ k[X, Y] is irreducible of total degree n with the coefficient of
Yⁿ equal to 1, then [F : k(x)] = n.
The monomials of degree less than n in the coordinates of a plane curve are
independent: let g ∈ k[X, Y] be irreducible of total degree n, with the coefficient of Yⁿ
equal to 1, and let g(x, y) = 0. Then the products x ^ i y ^ j with i + j < n are linearly
independent over k, since y has degree n over k(x).
Let g ∈ k[X, Y] be irreducible and monic in Y over k[X], and let y satisfy
g(x, y) = 0, where F / k(x) is separable. Then the partial derivative ∂g/∂Y does not vanish
at (x, y).
The different of a smooth plane curve above infinity: let g ∈ k[X, Y] be irreducible of
total degree n, with the coefficient of Yⁿ equal to 1, whose chart equation at infinity
h = uⁿ g(1 / u, v / u) has h, ∂h/∂u and ∂h/∂v generating the unit ideal, and let F / k(x)
be generated by y with g(x, y) = 0. At every place P of F at which x has no zero, in
particular at every pole of x, d(P ∣ P ∩ k(x)) = ord_P (∂g/∂Y (x, y)) - (n - 1) ord_P x.
The different of a smooth plane curve (Stichtenoth, Theorem 3.5.10, in both affine
charts): let g ∈ k[X, Y] be irreducible of total degree n, with the coefficient of Yⁿ equal
to 1, such that g = 0 is smooth and so is its chart at infinity h = uⁿ g(1 / u, v / u) = 0,
and let F / k(x) be finite separable, generated by y with g(x, y) = 0. Then
Diff(F / k(x)) = div (∂g/∂Y (x, y)) + (n - 1) (x)_∞.
The degree of the different of a smooth plane curve: under the hypotheses of
TauCeti.Divisor.different_eq_principal_add_zsmul_poles, deg Diff(F / k(x)) = n (n - 1) for a
curve of degree n.