# OO ToricDivisor -- make the associated rank-one reflexive sheaf

## Synopsis

• Operator: SPACE
• Usage:
OO D
• Inputs:
• D, ,
• Outputs:
• , the associated rank-one reflexive sheaf

## Description

For a Weil divisor $D$ on a normal variety $X$ the associated sheaf ${\cal O}_X(D)$ is defined by $H^0(U, {\cal O}_X(D)) = \{ f \in {\mathbb C}(X)^* | (div(f)+D)|_U \geq 0 \} \cup \{0\}$. The sheaf associated to a Weil divisor is reflexive; it is equal to its bidual. A divisor is Cartier if and only if the associated sheaf is a line bundle

The first examples show that the associated sheaves are reflexive.

 i1 : PP3 = toricProjectiveSpace 3; i2 : K = toricDivisor PP3 o2 = - PP3 - PP3 - PP3 - PP3 0 1 2 3 o2 : ToricDivisor on PP3 i3 : omega = OO K 1 o3 = OO (-4) PP3 o3 : coherent sheaf on PP3 i4 : omegaVee = prune sheafHom (omega, OO_PP3) 1 o4 = OO (4) PP3 o4 : coherent sheaf on PP3 i5 : omega === prune sheafHom (omegaVee, OO_PP3) o5 = true
 i6 : X = hirzebruchSurface 2; i7 : D = X_0 + X_1 o7 = X + X 0 1 o7 : ToricDivisor on X i8 : L = OO D 1 o8 = OO (-1, 1) X o8 : coherent sheaf on X i9 : LVee = prune sheafHom (L, OO_X) 1 o9 = OO (1, -1) X o9 : coherent sheaf on X i10 : L === prune sheafHom (LVee, OO_X) o10 = true
 i11 : rayList = {{1,0,0},{0,1,0},{0,0,1},{0,-1,-1},{-1,0,-1},{-2,-1,0}}; i12 : coneList = {{0,1,2},{0,1,3},{1,3,4},{1,2,4},{2,4,5},{0,2,5},{0,3,5},{3,4,5}}; i13 : Y = normalToricVariety(rayList,coneList); i14 : isSmooth Y o14 = false i15 : isProjective Y o15 = false i16 : E = Y_0 + Y_2 + Y_4 o16 = Y + Y + Y 0 2 4 o16 : ToricDivisor on Y i17 : isCartier E o17 = false i18 : F = OO E 1 o18 = OO (1, 3, 2) Y o18 : coherent sheaf on Y i19 : FVee = prune sheafHom(F, OO_Y) 1 o19 = OO (-1, -3, -2) Y o19 : coherent sheaf on Y i20 : F === prune sheafHom(FVee, OO_Y) o20 = true

Two Weil divisors $D$ and $E$ are linearly equivalent if $D = E + div(f)$, for some $f \in {\mathbb C}(X)^*$. Linearly equivalent divisors produce isomorphic sheaves.

 i21 : PP3 = toricProjectiveSpace 3; i22 : D1 = PP3_0 o22 = PP3 0 o22 : ToricDivisor on PP3 i23 : E1 = PP3_1 o23 = PP3 1 o23 : ToricDivisor on PP3 i24 : OO D1 === OO E1 o24 = true i25 : X = hirzebruchSurface 2; i26 : D2 = X_2 + X_3 o26 = X + X 2 3 o26 : ToricDivisor on X i27 : E2 = 3*X_0 + X_1 o27 = 3*X + X 0 1 o27 : ToricDivisor on X i28 : OO D2 === OO E2 o28 = true