Add Batyrev's construction of toric varieties to toric library

src/sage/schemes/toric/batyrev.py

@@ -0,0 +1,139 @@
+#!/usr/bin/env sage -python
+# IMPORTANT: Place this file outside the Sage source tree (e.g., in a separate project folder).
+# Run with Sage's Python: `sage -python batyrev.py` or import in a Sage session after cd into the directory.
+import sys
+if 'sage' not in sys.executable.lower():
+    sys.exit("Error: run with Sage's Python (e.g. `sage -python batyrev.py`).")
+import sys
+# Ensure running under Sage Python
+if 'sage' not in sys.executable.lower():
+    sys.exit("Error: Please run this script using Sage's Python (e.g. via `./batyrev.py` or `sage -python batyrev.py`).")
+
+r"""
+Library for Batyrev's construction of smooth complete toric varieties of Picard rank 3. [Batyrev1991]
+TODO: Batyrev's construction for 18 toric Fano 4-folds [Batyrev1999]
+TODO: Batyrev's construction for 123 toric Fano 4-folds [Batyrev1999]
+"""
+
+
+from sage.structure.sage_object import SageObject
+
+from sage.matrix.constructor import matrix
+from sage.matrix.special import identity_matrix
+from sage.geometry.fan import Fan
+from sage.geometry.lattice_polytope import LatticePolytope
+from sage.geometry.toric_lattice import ToricLattice
+from sage.rings.integer_ring import ZZ
+from sage.rings.rational_field import QQ
+from sage.arith.misc import GCD as gcd
+from sage.schemes.toric.variety import (DEFAULT_PREFIX,
+                                        ToricVariety,
+                                        normalize_names)
+
+# from sage.schemes.toric.fano_variety import CPRFanoToricVariety AL: so far, seems not used in this work
+
+
+
+class BatyrevToricFactory(SageObject):
+    r"""
+    The methods of this class construct Batyrev' classification of smooth and complete toric varieties.`
+    """
+
+    _check = True # AL: what does this do?
+
+    def __init__(self):
+        self.toric_varieties_rays_cones = {}
+
+    def _make_ToricVariety(self, name, coordinate_names, base_ring):
+        r"""
+        Construct a toric variety and cache the result.
+
+        INPUT:
+
+        - ``name`` -- string; one of the pre-defined names in the
+          ``toric_varieties_rays_cones`` data structure
+
+        - ``coordinate_names`` -- string describing the names of the
+          homogeneous coordinates of the toric variety
+
+        - ``base_ring`` -- a ring (default: `\QQ`); the base ring for
+          the toric variety
+
+        OUTPUT: a :class:`toric variety
+        <sage.schemes.toric.variety.ToricVariety_field>`.
+
+        EXAMPLES::
+
+            sage: toric_varieties.A1()           # indirect doctest
+            1-d affine toric variety
+        """
+        rays, cones = self.toric_varieties_rays_cones[name]
+        if coordinate_names is None:
+            dict_key = (name, base_ring)
+        else:
+            coordinate_names = normalize_names(coordinate_names, len(rays),
+                                               DEFAULT_PREFIX)
+            dict_key = (name, base_ring) + tuple(coordinate_names)
+        if dict_key not in self.__dict__:
+            fan = Fan(cones, rays, check=self._check)
+            self.__dict__[dict_key] = \
+                ToricVariety(fan,
+                             coordinate_names=coordinate_names,
+                             base_ring=base_ring)
+        return self.__dict__[dict_key]
+
+    def _make_batyrev_matrix(self, P, B, C):
+        p0, p1, p2, p3, p4 = P
+        Cprime = [0] + list(C)
+        R1 = ([1]*p0) + ([1]*p1) + ([-c for c in Cprime]) + ([-(b+1) for b in B]) + ([0]*p4)
+        R2 = ([0]*p0) + ([1]*p1) + ([1]*p2) + ([0]*p3) + ([-1]*p4)
+        R3 = ([0]*p0) + ([0]*p1) + ([1]*p2) + ([1]*p3) + ([0]*p4)
+        R4 = ([0]*p0) + ([-1]*p1) + ([0]*p2) + ([1]*p3) + ([1]*p4)
+        R5 = ([1]*p0) + ([0]*p1) + ([-c for c in Cprime]) + ([-b for b in B]) + ([1]*p4)
+        return matrix(ZZ, [R1, R2, R3, R4, R5])
+
+    def add_family(self, name, P, B, C):
+        """
+        Register a family by its Batyrev parameters P, B, C.
+        P = [p0, p1, p2, p3, p4]
+        B is a list of length p3
+        C is a list of length p2 - 1
+        """
+        A = self._make_batyrev_matrix(P, B, C)
+        K = A.right_kernel()
+        rays = [v for v in K.basis()]
+        p0, p1, p2, p3, p4 = P
+        X0 = list(range(p0))
+        X1 = list(range(p0, p0+p1))
+        X2 = list(range(p0+p1, p0+p1+p2))
+        X3 = list(range(p0+p1+p2, p0+p1+p2+p3))
+        X4 = list(range(p0+p1+p2+p3, p0+p1+p2+p3+p4))
+        X5 = X0 + X4
+        primitive_cols = [X0, X1, X2, X3, X5]
+        d = len(rays[0])
+        cones = []
+        from itertools import combinations
+        for combo in combinations(range(len(rays)), d):
+            if any(set(pc) <= set(combo) for pc in primitive_cols):
+                continue
+            cones.append(list(combo))
+        self.toric_varieties_rays_cones[name] = (rays, cones)
+
+    def make_variety(self, name, P, B, C, coordinate_names=None, base_ring=QQ):
+        """
+        Convenience method: add the family and return its ToricVariety.
+        """
+        self.add_family(name, P, B, C)
+        return self._make_ToricVariety(name, coordinate_names, base_ring)
+
+# # instantiate and register example families
+# btf = BatyrevToricFactory()
+# # rank-3 Fano 3-folds examples
+# btf.add_family('Batyrev_3_26', [2,1,1,1,1], [-2], [])
+# btf.add_family('Batyrev_3_29', [1,1,1,1,1], [2], [])
+
+
+if __name__ == '__main__':
+    # simple test
+    X = btf.make_variety('Test', [2,1,1,1,1], [-2], [])
+    print('smooth?', X.is_smooth(), 'rho=', X.picard_number(), 'dim=', X.dimension())

src/sage/schemes/toric/meson.build

@@ -1,6 +1,7 @@
 py.install_sources(
   '__init__.py',
   'all.py',
+  'batyrev.py',
   'chow_group.py',
   'divisor.py',
   'fano_variety.py',