Add sheaf cohomology for projective schemes

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

@@ -4049,6 +4049,10 @@ REFERENCES:
              293-311 (1987).
              :doi:`10.1007/BF00337892`
 
+.. [Kudo2017] Momonari Kudo, "Analysis of an algorithm to compute the cohomology
+              groups of coherent sheaves and its applications", Japan Journal of
+              Industrial and Applied Mathematics 34 (2017), 1--40.
+
 .. [Kuh1987] \W. Kühnel, "Minimal triangulations of Kummer varieties",
              Abh. Math. Sem. Univ. Hamburg 57 (1987), 7-20.
 

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

@@ -46,6 +46,15 @@ Projective Schemes
    sage/schemes/projective/projective_rational_point
    sage/schemes/projective/projective_homset
 
+Coherent Sheaves
+----------------
+
+.. toctree::
+   :maxdepth: 1
+
+   sage/schemes/sheaves/sheaf
+   sage/schemes/sheaves/cohomology
+
 Products of Projective Spaces
 -----------------------------
 
@@ -61,6 +70,7 @@ Products of Projective Spaces
 
 Toric Varieties
 ---------------
+
 .. toctree::
    :maxdepth: 1
 
@@ -85,6 +95,7 @@ Toric Varieties
 
 Cyclic Covers
 ---------------
+
 .. toctree::
    :maxdepth: 1
 
@@ -95,6 +106,7 @@ Cyclic Covers
 
 Berkovich Analytic Space
 ------------------------
+
 .. toctree::
    :maxdepth: 1
 

src/sage/misc/latex.py

@@ -583,6 +583,10 @@ def latex_extra_preamble():
         \newcommand{\PSL}{\mathrm{PSL}}
         \newcommand{\lcm}{\mathop{\operatorname{lcm}}}
         \newcommand{\dist}{\mathrm{dist}}
+        \newcommand{\im}{\mathop{\operatorname{im}}}
+        \newcommand{\rank}{\mathop{\operatorname{rank}}}
+        \newcommand{\PP}{\mathbf{P}}
+        \newcommand{\OO}{\mathcal{O}}
         \newcommand{\Bold}[1]{\mathbf{#1}}
         <BLANKLINE>
     """

src/sage/misc/latex_macros.py

@@ -175,7 +175,12 @@ def convert_latex_macro_to_mathjax(macro):
 latex_macros = [r"\newcommand{\SL}{\mathrm{SL}}",
                 r"\newcommand{\PSL}{\mathrm{PSL}}",
                 r"\newcommand{\lcm}{\mathop{\operatorname{lcm}}}",
-                r"\newcommand{\dist}{\mathrm{dist}}"]
+                r"\newcommand{\dist}{\mathrm{dist}}",
+                r"\newcommand{\im}{\mathop{\operatorname{im}}}",
+                r"\newcommand{\rank}{\mathop{\operatorname{rank}}}",
+                r"\newcommand{\PP}{\mathbf{P}}",
+                r"\newcommand{\OO}{\mathcal{O}}",
+                ]
 
 # The following is to allow customization of typesetting of rings:
 # mathbf vs mathbb.  See latex.py for more information.

src/sage/schemes/affine/affine_point.py

@@ -1,27 +1,19 @@
 r"""
 Points on affine varieties
 
-Scheme morphism for points on affine varieties.
-
+This module implements scheme morphism for points on affine varieties.
 
 AUTHORS:
 
-- David Kohel, William Stein
-
-- Volker Braun (2011-08-08): Renamed classes, more documentation, misc
-  cleanups.
-
-- Ben Hutz (2013)
+- David Kohel, William Stein (2006): initial version
+- Volker Braun (2011-08-08): renamed classes, more documentation, misc cleanups
+- Ben Hutz (2013): many improvements
 """
 
-# Historical note: in trac #11599, V.B. renamed
-# * _point_morphism_class -> _morphism
-# * _homset_class -> _point_homset
-
 # ****************************************************************************
-#       Copyright (C) 2011 Volker Braun <[email protected]>
 #       Copyright (C) 2006 David Kohel <[email protected]>
 #       Copyright (C) 2006 William Stein <[email protected]>
+#       Copyright (C) 2011 Volker Braun <[email protected]>
 #
 #  Distributed under the terms of the GNU General Public License (GPL)
 #  as published by the Free Software Foundation; either version 2 of
@@ -37,10 +29,10 @@
 _NumberFields = NumberFields()
 
 
-############################################################################
-# Rational points on schemes, which we view as morphisms determined
-# by coordinates.
-############################################################################
+# --------------------------------------------------------------------
+# Rational points on schemes, which we view as morphisms determined by
+# coordinates.
+# --------------------------------------------------------------------
 
 class SchemeMorphism_point_affine(SchemeMorphism_point):
     """
@@ -410,6 +402,29 @@ def multiplicity(self):
             raise TypeError("this point must be a point on an affine subscheme")
         return self.codomain().multiplicity(self)
 
+    def as_subscheme(self):
+        r"""
+        Return the subscheme associated with this rational point.
+
+        EXAMPLES::
+
+            sage: A2.<x,y> = AffineSpace(QQ, 2)
+            sage: p1 = A2.point([0,0]).as_subscheme(); p1
+            Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
+              x, y
+            sage: p2 = A2.point([1,1]).as_subscheme(); p2
+            Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
+              x - 1, y - 1
+            sage: p1 + p2
+            Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
+              x - y, y^2 - y
+        """
+        A = self.codomain().ambient_space()
+        g = A.gens()
+        v = self._coords
+        n = len(v)
+        return A.subscheme([g[i] - v[i] for i in range(n)])
+
 
 class SchemeMorphism_point_affine_finite_field(SchemeMorphism_point_affine_field):
 

src/sage/schemes/affine/affine_space.py

@@ -1,5 +1,5 @@
 """
-Affine `n` space over a ring
+Affine `n`-space over a ring
 """
 
 # ****************************************************************************

src/sage/schemes/projective/projective_point.py

@@ -1,29 +1,23 @@
 r"""
 Points on projective varieties
 
-Scheme morphism for points on projective varieties
-
-
+This module implements scheme morphism for points on projective varieties.
 
 AUTHORS:
 
-- David Kohel, William Stein
-
-- William Stein (2006-02-11): fixed bug where P(0,0,0) was allowed as
-  a projective point.
-
-- Volker Braun (2011-08-08): Renamed classes, more documentation, misc
-  cleanups.
-
-- Ben Hutz (June 2012) added support for projective ring;
-  (March 2013) iteration functionality and new directory structure
+- David Kohel, William Stein (2006): initial version
+- William Stein (2006-02-11): fixed bug where P(0,0,0) was allowed as a
+  projective point
+- Volker Braun (2011-08-08): Renamed classes, more documentation, misc cleanups
+- Ben Hutz (2012-06): added support for projective ring
+- Ben Hutz (2013-03): added iteration functionality and new directory structure
   for affine/projective, height functionality
 """
 
 # ****************************************************************************
-#       Copyright (C) 2011 Volker Braun <[email protected]>
 #       Copyright (C) 2006 David Kohel <[email protected]>
 #       Copyright (C) 2006 William Stein <[email protected]>
+#       Copyright (C) 2011 Volker Braun <[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
@@ -52,9 +46,11 @@
 from sage.structure.sequence import Sequence
 from sage.structure.richcmp import richcmp, op_EQ, op_NE
 
-#*******************************************************************
+
+# --------------------
 # Projective varieties
-#*******************************************************************
+# --------------------
+
 class SchemeMorphism_point_projective_ring(SchemeMorphism_point):
     """
     A rational point of projective space over a ring.
@@ -1054,6 +1050,7 @@ def is_preperiodic(self, f, err=0.1, return_period=False):
         except AttributeError:
             raise TypeError("map must be a dynamical system")
 
+
 class SchemeMorphism_point_projective_field(SchemeMorphism_point_projective_ring):
     """
     A rational point of projective space over a field.
@@ -1401,6 +1398,35 @@ def multiplicity(self):
             raise TypeError("this point must be a point on a projective subscheme")
         return self.codomain().multiplicity(self)
 
+    def as_subscheme(self):
+        r"""
+        Return the subscheme associated with this rational point.
+
+        EXAMPLES::
+
+            sage: P2.<x,y,z> = ProjectiveSpace(QQ,2)
+            sage: p1 = P2.point([0,0,1]).as_subscheme(); p1
+            Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
+              x, y
+            sage: p2 = P2.point([1,1,1]).as_subscheme(); p2
+            Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
+              x - z, y - z
+            sage: p1 + p2
+            Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
+              x - y, y^2 - y*z
+        """
+        P = self.codomain().ambient_space()
+        g = P.gens()
+        v = self._coords
+        n = len(v)
+        for i in range(n - 1, -1, -1):
+            if v[i]:
+                break
+        a = v[i]
+        x = g[i]
+        return P.subscheme([a*g[j] - v[j]*x for j in range(n) if j != i])
+
+
 class SchemeMorphism_point_projective_finite_field(SchemeMorphism_point_projective_field):
 
     def __hash__(self):
@@ -1438,9 +1464,11 @@ def __hash__(self):
         N = self.codomain().ambient_space().dimension_relative()
         return hash(sum(hash(self[i]) * p**i for i in range(N + 1)))
 
-#*******************************************************************
+
+# -----------------
 # Abelian varieties
-#*******************************************************************
+# -----------------
+
 class SchemeMorphism_point_abelian_variety_field(AdditiveGroupElement, SchemeMorphism_point_projective_field):
     """
     A rational point of an abelian variety over a field.

src/sage/schemes/projective/projective_space.py

@@ -1,5 +1,5 @@
 r"""
-Projective `n` space over a ring
+Projective `n`-space over a ring
 
 EXAMPLES:
 
@@ -110,6 +110,7 @@
 from sage.combinat.tuple import UnorderedTuples
 from sage.combinat.subset import Subsets
 from sage.matrix.constructor import matrix
+from sage.modules.free_module import FreeModule
 from sage.modules.free_module_element import prepare
 from sage.schemes.generic.ambient_space import AmbientSpace
 from sage.schemes.projective.projective_homset import (SchemeHomset_points_projective_ring,
@@ -2266,6 +2267,71 @@ def line_through(self, p, q):
         m = matrix(3, list(self.gens()) + list(p) + list(q))
         return Curve([f for f in m.minors(3) if f])
 
+    def coherent_sheaf(self, module, twist=0):
+        r"""
+        Return the sheaf defined by the graded ``module``.
+
+        If ``twist`` is a non-zero integer `n`, the sheaf twisted by
+        `\OO_{\PP^r}(n)` is returned.
+
+        INPUT:
+
+        - ``module`` -- a free module or a quotient module over the coordinate
+          ring of this space
+
+        - ``twist`` -- (default: `0`) an integer
+
+        EXAMPLES::
+
+            sage: P2 = ProjectiveSpace(QQ, 2, 'x')
+            sage: S = P2.coordinate_ring()
+            sage: SS = FreeModule(S, 2)
+            sage: P2.coherent_sheaf(SS, twist=2)
+            Twisted Sheaf on Projective Space of dimension 2 over Rational Field
+        """
+        from sage.schemes.sheaves.sheaf import Sheaf_on_projective_space
+        return Sheaf_on_projective_space(self, module, twist=twist)
+
+    def structure_sheaf(self, twist=0):
+        r"""
+        Return the structure sheaf `\OO_{\PP^r}` of this projective space.
+
+        If ``twist`` is a non-zero integer `n`, the sheaf twisted by
+        `\OO_{\PP^r}(n)` is returned.
+
+        INPUT:
+
+        - ``twist`` -- (default: `0`) an integer
+
+        EXAMPLES::
+
+            sage: P3.<x0,x1,x2,x3> = ProjectiveSpace(QQ, 3)
+            sage: P3.structure_sheaf()
+            Sheaf on Projective Space of dimension 3 over Rational Field
+        """
+        M = FreeModule(self.coordinate_ring(), rank=1)
+        return self.coherent_sheaf(M, twist=twist)
+
+    def arithmetic_genus(self):
+        r"""
+        Return the arithmetic genus of this projective space.
+
+        This is known to be zero. This method simply checks the fact.
+
+        EXAMPLES::
+
+            sage: P1 = ProjectiveSpace(QQ, 1)
+            sage: P1.arithmetic_genus()
+            0
+            sage: P2 = ProjectiveSpace(QQ, 2)
+            sage: P2.arithmetic_genus()
+            0
+        """
+        p = self.structure_sheaf().euler_characteristic() - 1
+        if self.dimension() % 2:
+            p = -p
+        return p
+
 
 class ProjectiveSpace_finite_field(ProjectiveSpace_field):
     def _point(self, *args, **kwds):