Basic functionalities for weighted projective {curves, points, spaces}

.gitignore

@@ -243,6 +243,7 @@ src/sage/schemes/hyperelliptic_curves/__init__.py
 src/sage/schemes/berkovich/__init__.py
 src/sage/schemes/generic/__init__.py
 src/sage/schemes/projective/__init__.py
+src/sage/schemes/weighted_projective/__init__.py
 src/sage/schemes/__init__.py
 src/sage/schemes/affine/__init__.py
 src/sage/modular/hecke/__init__.py

src/doc/en/reference/curves/index.rst

@@ -15,6 +15,7 @@ Curves
    sage/schemes/curves/affine_curve
    sage/schemes/curves/plane_curve_arrangement
    sage/schemes/curves/projective_curve
+   sage/schemes/curves/weighted_projective_curve
    sage/schemes/curves/point
    sage/schemes/curves/closed_point
    sage/schemes/curves/zariski_vankampen

src/doc/en/reference/schemes/index.rst

@@ -47,6 +47,16 @@ Projective Schemes
    sage/schemes/projective/projective_homset
    sage/schemes/projective/proj_bdd_height
 
+Weighted Projective Schemes
+---------------------------
+
+.. toctree::
+   :maxdepth: 1
+
+   sage/schemes/weighted_projective/weighted_projective_space
+   sage/schemes/weighted_projective/weighted_projective_point
+   sage/schemes/weighted_projective/weighted_projective_homset
+
 Products of Projective Spaces
 -----------------------------
 

src/sage/schemes/all.py

@@ -42,6 +42,8 @@
 
 from sage.schemes.product_projective.all import *
 
+from sage.schemes.weighted_projective.all import *
+
 from sage.schemes.cyclic_covers.all import *
 
 from sage.schemes.berkovich.all import *

src/sage/schemes/curves/constructor.py

@@ -18,6 +18,12 @@
     Projective Plane Curve over Finite Field of size 5
      defined by -x^9 + y^2*z^7 - x*z^8
 
+Here, we construct a hyperelliptic curve manually::
+
+    sage: WP.<x,y,z> = WeightedProjectiveSpace([1, 3, 1], GF(103))
+    sage: Curve(y^2 - (x^5*z + 17*x^2*z^4 + 92*z^6), WP)
+    Weighted Projective Curve over Finite Field of size 103 defined by y^2 - x^5*z - 17*x^2*z^4 + 11*z^6
+
 AUTHORS:
 
 - William Stein (2005-11-13)
@@ -49,6 +55,7 @@
 from sage.schemes.generic.algebraic_scheme import AlgebraicScheme
 from sage.schemes.affine.affine_space import AffineSpace, AffineSpace_generic
 from sage.schemes.projective.projective_space import ProjectiveSpace, ProjectiveSpace_ring
+from sage.schemes.weighted_projective.weighted_projective_space import WeightedProjectiveSpace_ring
 from sage.schemes.plane_conics.constructor import Conic
 
 from .projective_curve import (ProjectiveCurve,
@@ -71,6 +78,8 @@
                            IntegralAffinePlaneCurve,
                            IntegralAffinePlaneCurve_finite_field)
 
+from .weighted_projective_curve import WeightedProjectiveCurve
+
 
 def _is_irreducible_and_reduced(F) -> bool:
     """
@@ -376,5 +385,12 @@ def Curve(F, A=None):
             return ProjectivePlaneCurve_field(A, F)
         return ProjectivePlaneCurve(A, F)
 
+    elif isinstance(A, WeightedProjectiveSpace_ring):
+        # currently, we only support curves in a weighted projective plane
+        if n != 2:
+            raise NotImplementedError("ambient space has to be a weighted projective plane")
+        # currently, we do not perform checks on weighted projective curves
+        return WeightedProjectiveCurve(A, F)
+
     else:
         raise TypeError('ambient space neither affine nor projective')

src/sage/schemes/curves/weighted_projective_curve.py

@@ -0,0 +1,105 @@
+# sage.doctest: needs sage.libs.singular
+r"""
+Weighted projective curves
+
+Weighted projective curves in Sage are curves in a weighted projective space or
+a weighted projective plane.
+
+EXAMPLES:
+
+For now, only curves in weighted projective plane is supported::
+
+    sage: WP.<x, y, z> = WeightedProjectiveSpace([1, 3, 1], QQ)
+    sage: C1 = WP.curve(y^2 - x^5 * z - 3 * x^2 * z^4 - 2 * z^6); C1
+    Weighted Projective Curve over Rational Field defined by y^2 - x^5*z - 3*x^2*z^4 - 2*z^6
+    sage: C2 = Curve(y^2 - x^5 * z - 3 * x^2 * z^4 - 2 * z^6, WP); C2
+    Weighted Projective Curve over Rational Field defined by y^2 - x^5*z - 3*x^2*z^4 - 2*z^6
+    sage: C1 == C2
+    True
+
+AUTHORS:
+
+- Gareth Ma (2025)
+"""
+
+# ****************************************************************************
+#  Copyright (C) 2005 William Stein <[email protected]>
+#  Copyright (C) 2025 Gareth Ma <[email protected]>
+#
+# This program is free software: you can redistribute it and/or modify
+# it under the terms of the GNU General Public License as published by
+# the Free Software Foundation, either version 2 of the License, or
+# (at your option) any later version.
+#                  https://www.gnu.org/licenses/
+# ****************************************************************************
+
+from sage.schemes.curves.curve import Curve_generic
+from sage.schemes.weighted_projective.weighted_projective_space import WeightedProjectiveSpace_ring
+
+
+class WeightedProjectiveCurve(Curve_generic):
+    """
+    Curves in weighted projective spaces.
+
+    EXAMPLES:
+
+    We construct a hyperelliptic curve manually::
+
+        sage: WP.<x, y, z> = WeightedProjectiveSpace([1, 3, 1], QQ)
+        sage: C = Curve(y^2 - x^5 * z - 3 * x^2 * z^4 - 2 * z^6, WP); C
+        Weighted Projective Curve over Rational Field defined by y^2 - x^5*z - 3*x^2*z^4 - 2*z^6
+    """
+    def __init__(self, A, X, *kwargs):
+        if not isinstance(A, WeightedProjectiveSpace_ring):
+            raise TypeError(f"A(={A}) is not a weighted projective space")
+        super().__init__(A, X, *kwargs)
+
+    def _repr_type(self) -> str:
+        r"""
+        Return a string representation of the type of this curve.
+
+        EXAMPLES::
+
+            sage: WP.<x,y,z> = WeightedProjectiveSpace([1, 3, 1], QQ)
+            sage: C = Curve(y^2 - x^5 * z - 3 * x^2 * z^4 - 2 * z^6, WP); C
+            Weighted Projective Curve over Rational Field defined by y^2 - x^5*z - 3*x^2*z^4 - 2*z^6
+            sage: C._repr_type()
+            'Weighted Projective'
+        """
+        return "Weighted Projective"
+
+    def projective_curve(self):
+        r"""
+        Return this weighted projective curve as a projective curve.
+
+        A weighted homogeneous polynomial `f(x_1, \ldots, x_n)`, where `x_i` has
+        weight `w_i`, can be viewed as an unweighted homogeneous polynomial
+        `f(y_1^{w_1}, \ldots, y_n^{w_n})`. This correspondence extends to
+        varieties.
+
+        .. TODO:
+
+            Implement homsets for weighted projective spaces and implement this
+            as a ``projective_embedding`` method instead.
+
+        EXAMPLES::
+
+            sage: WP = WeightedProjectiveSpace([1, 3, 1], QQ, "x, y, z")
+            sage: x, y, z = WP.gens()
+            sage: C = WP.curve(y^2 - (x^5*z + 3*x^2*z^4 - 2*x*z^5 + 4*z^6)); C
+            Weighted Projective Curve over Rational Field defined by y^2 - x^5*z - 3*x^2*z^4 + 2*x*z^5 - 4*z^6
+            sage: C.projective_curve()
+            Projective Plane Curve over Rational Field defined by y^6 - x^5*z - 3*x^2*z^4 + 2*x*z^5 - 4*z^6
+        """
+        from sage.schemes.projective.projective_space import ProjectiveSpace
+
+        WP = self.ambient_space()
+        PP = ProjectiveSpace(WP.dimension_relative(), WP.base_ring(), WP.variable_names())
+        PP_ring = PP.coordinate_ring()
+        subs_dict = {name: var**weight for (name, var), weight in
+                     zip(WP.gens_dict().items(), WP.weights())}
+
+        wp_polys = self.defining_polynomials()
+        pp_polys = [PP_ring(poly.subs(**subs_dict)) for poly in wp_polys]
+
+        return PP.curve(pp_polys)

src/sage/schemes/meson.build

@@ -18,4 +18,5 @@ install_subdir('plane_quartics', install_dir: sage_install_dir / 'schemes')
 install_subdir('product_projective', install_dir: sage_install_dir / 'schemes')
 install_subdir('projective', install_dir: sage_install_dir / 'schemes')
 install_subdir('riemann_surfaces', install_dir: sage_install_dir / 'schemes')
+install_subdir('weighted_projective', install_dir: sage_install_dir / 'schemes')
 subdir('toric')

src/sage/schemes/weighted_projective/all.py

@@ -0,0 +1,16 @@
+"""nodoctest
+all.py -- export of projective schemes to Sage
+"""
+
+# ****************************************************************************
+#  Copyright (C) 2005 William Stein <[email protected]>
+#  Copyright (C) 2025 Gareth Ma <[email protected]>
+#
+# This program is free software: you can redistribute it and/or modify
+# it under the terms of the GNU General Public License as published by
+# the Free Software Foundation, either version 2 of the License, or
+# (at your option) any later version.
+#                  https://www.gnu.org/licenses/
+# ****************************************************************************
+
+from sage.schemes.weighted_projective.weighted_projective_space import WeightedProjectiveSpace

src/sage/schemes/weighted_projective/weighted_projective_homset.py

@@ -0,0 +1,38 @@
+"""
+Hom-sets of weighted projective schemes
+
+AUTHORS:
+
+- Gareth Ma (2024): initial version, based on unweighted version.
+"""
+
+# *****************************************************************************
+#        Copyright (C) 2006 William Stein <[email protected]>
+#        Copyright (C) 2011 Volker Braun <[email protected]>
+#        Copyright (C) 2024 Gareth Ma <[email protected]>
+#
+#   Distributed under the terms of the GNU General Public License (GPL)
+#   as published by the Free Software Foundation; either version 2 of
+#   the License, or (at your option) any later version.
+#                   http://www.gnu.org/licenses/
+# *****************************************************************************
+
+from sage.schemes.generic.homset import SchemeHomset_points
+
+
+class SchemeHomset_points_weighted_projective_ring(SchemeHomset_points):
+    """
+    Set of rational points of a weighted projective variety over a ring.
+
+    INPUT:
+
+    See :class:`SchemeHomset_points`.
+
+    EXAMPLES::
+
+        sage: W = WeightedProjectiveSpace([3, 4, 5], QQ)
+        sage: W.point_homset()
+        Set of rational points of Weighted Projective Space of dimension 2 with weights (3, 4, 5) over Rational Field
+        sage: W.an_element().parent() is W.point_homset()
+        True
+    """