The canonical divisor and the regular differentials of a smooth plane curve #
Let g ∈ k[X, Y] be irreducible of total degree n, with the coefficient of Yⁿ equal to 1,
such that the projective closure of g = 0 is smooth in both affine charts g = 0 and
uⁿ g(1 / u, v / u) = 0, and let F / k(x) be finite separable, generated by y with
g(x, y) = 0, with exact constant field k. The divisor of dx is
-2 (x)_∞ + Diff(F / k(x)), and Diff(F / k(x)) = div (∂g/∂Y (x, y)) + (n - 1) (x)_∞
(TauCeti.Divisor.different_eq_principal_add_zsmul_poles), so the Weil differential
ω₀ = dx / (∂g/∂Y (x, y))
has divisor (n - 3) (x)_∞. Since the projective closure does not pass through (0 : 1 : 0),
the pole divisor (x)_∞ is the divisor cut out on the curve by the line at infinity, and this says
that the effective canonical divisors of a smooth plane curve of degree n are cut out by the
curves of
degree n - 3.
The monomials x ^ i y ^ j with i + j ≤ n - 3 lie in L((n - 3) (x)_∞): y is integral over
k[x], and y / x is integral over k[1 / x] by the equation of the chart at infinity, so y
has poles only where x does, and of no larger order. They are linearly independent, because
1, y, …, y ^ (n - 1) is a basis of F / k(x), and there are (n - 1)(n - 2) / 2 = g of them,
while ℓ((n - 3) (x)_∞) = g since the divisor is canonical. Hence L((n - 3) (x)_∞) is exactly
the space of polynomials in x and y of degree at most n - 3, and the regular differentials of
F are exactly the
h(x, y) dx / (∂g/∂Y (x, y)) with deg h ≤ n - 3.
Main results #
TauCeti.finrank_span_pow_mul_pow_eq_genusandTauCeti.riemannRochSpace_eq_span_pow_mul_pow_of_divisorClass_eq_canonicalClass: if the monomialsx ^ i y ^ jwithi + j ≤ n - 3are linearly independent and2g = (n - 1)(n - 2), they span a space of dimensiong, and hence every canonicalL(D)containing them.TauCeti.mem_riemannRochSpace_poles_of_aeval_eq_zero:y ∈ L((x)_∞).TauCeti.linearIndepOn_pow_mul_pow_of_aeval_eq_zero: the monomialsx ^ i y ^ jwithi + j < nare linearly independent.TauCeti.weilDifferentialDivisor_repartitionDualMul_inv_weilDifferentialCotrace_of_aeval_eq_zero: the differentialdx / (∂g/∂Y (x, y))has divisor(n - 3) (x)_∞.TauCeti.divisorClass_sub_three_zsmul_poles_eq_canonicalClass_of_aeval_eq_zero:(n - 3) (x)_∞is canonical.TauCeti.riemannRochSpace_sub_three_zsmul_poles_eq_span_of_aeval_eq_zero:L((n - 3) (x)_∞)is spanned by the monomialsx ^ i y ^ jwithi + j ≤ n - 3.TauCeti.weilDifferentialFiltration_zero_eq_span_of_aeval_eq_zero: the regular differentials are spanned by thex ^ i y ^ j dx / (∂g/∂Y (x, y))withi + j ≤ n - 3.
References #
- W. Fulton, Algebraic Curves, Section 8.5, for the canonical divisors of a plane curve and the
adjoint curves of degree
n - 3. - H. Stichtenoth, Algebraic Function Fields and Codes, 2nd ed., GTM 254, Springer, 2009, Remark 4.3.7(c) and Theorem 3.5.10.
TauCeti.FieldTheory.FunctionField.Hyperelliptic.RegularDifferential, the corresponding description of the regular differentials ofy² = f(x), whose proof this file follows.
The monomials of degree at most n - 3 span a space of dimension g: if the products
x ^ i y ^ j with i + j ≤ n - 3 are linearly independent, n ≥ 1 and 2g = (n - 1)(n - 2),
then they span a space of dimension g, since there are (n - 1)(n - 2) / 2 of them.
A canonical Riemann–Roch space spanned by monomials: under the hypotheses of
TauCeti.finrank_span_pow_mul_pow_eq_genus, if the monomials x ^ i y ^ j with i + j ≤ n - 3
lie in L(D) for a divisor D in the canonical class, then they span L(D), since
ℓ(D) = g.
The coordinate y of a plane curve has no worse poles than x: let g ∈ k[X, Y] have
total degree n, with the coefficient of Yⁿ equal to 1, and let g(x, y) = 0. Then
y ∈ L((x)_∞). Indeed y is integral over k[x], and y / x is integral over k[1 / x]
through the equation uⁿ g(1 / u, v / u) = 0 of the chart at infinity.
Monomials of degree d in the coordinates of a plane curve lie in L(d (x)_∞): under the
hypotheses of TauCeti.mem_riemannRochSpace_poles_of_aeval_eq_zero, x ^ i y ^ j ∈ L(m (x)_∞)
whenever i + j ≤ m.
The divisor of dx / (∂g/∂Y (x, y)): 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 uⁿ g(1 / u, v / u) = 0, and let F / k(x) be finite separable, generated by y with
g(x, y) = 0, with exact constant field k. Then the Weil differential dx / (∂g/∂Y (x, y)),
where dx is the cotrace of the normalized differential of k(x), has divisor (n - 3) (x)_∞.
The effective canonical divisors of a smooth plane curve of degree n are cut out by
curves of degree
n - 3: under the hypotheses of
TauCeti.weilDifferentialDivisor_repartitionDualMul_inv_weilDifferentialCotrace_of_aeval_eq_zero,
the divisor (n - 3) (x)_∞ represents the canonical class.
The Riemann–Roch space of the canonical divisor of a smooth plane curve: under the
hypotheses of
TauCeti.weilDifferentialDivisor_repartitionDualMul_inv_weilDifferentialCotrace_of_aeval_eq_zero,
L((n - 3) (x)_∞) is spanned by the monomials x ^ i y ^ j with i + j ≤ n - 3. These are
linearly independent (Transcendental.linearIndepOn_pow_mul_pow), so they form a basis, and
L((n - 3) (x)_∞) is the space of polynomials in x and y of degree at most n - 3.
The regular differentials of a smooth plane curve: under the hypotheses of
TauCeti.weilDifferentialDivisor_repartitionDualMul_inv_weilDifferentialCotrace_of_aeval_eq_zero,
the regular Weil differentials of F are spanned by the x ^ i y ^ j dx / (∂g/∂Y (x, y)) with
i + j ≤ n - 3, where dx is the cotrace of the normalized differential of k(x): they are the
h(x, y) dx / (∂g/∂Y (x, y)) for the polynomials h of degree at most n - 3.