Algebraic geometry codes from polyhedral divisors

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Journal of Symbolic Computation

journal homepage:www.elsevier.com/locate/jsc

Algebraic geometry codes from polyhedral divisors

Nathan Owen Ilten

a

, Hendrik Süß

b,1

a_{Mathematisches Institut, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany}

b_{Mathematisches Institut, Brandenburgische Technische Universität Cottbus, PF 10 13 44, 03013 Cottbus, Germany}

a r t i c l e i n f o

Article history:

Received 5 November 2008 Accepted 6 July 2009 Available online 25 March 2010

Keywords:

AG codes Evaluation codes Toric varieties

a b s t r a c t

A description of complete normal varieties with lower-dimensional torus action has been given byAltmann et al.(2008), generalizing the theory of toric varieties. Considering the case where the acting torusThas codimension one, we describeT-invariant Weil and Cartier divisors and provide formulae for calculating global sections, intersection numbers, and Euler characteristics. As an application, we use divisors on these so-calledT-varieties to define new evaluation codes calledT-codes. We find estimates on their minimum distance using intersection theory. This generalizes the theory of toric codes and combines it with AG codes on curves. As the simplest application of our general techniques we look at codes on ruled surfaces coming from decomposable vector bundles. Already this construction gives codes that are better than the related product code. Further examples show that we can improve these codes by constructing more sophisticatedT-varieties. These results suggest looking further for good codes onT-varieties.

©2010 Elsevier Ltd. All rights reserved.

1. Introduction

An important class of linear codes is the class of algebraic geometry Codes, introduced byGoppa (1981). These codes arise by evaluating global sections of a line bundle on a curve overFqat a number

of_Fq-rational points; good estimates on the dimension and minimum distance of such codes can be

obtained by using the theorem of Riemann and Roch. Such codes have been generalized to higher-dimensional varieties. It is however often difficult to obtain non-trivial estimates on the parameters of

E-mail addresses:[email protected](N.O. Ilten),[email protected](H. Süß).

URL:http://people.cs.uchicago.edu/∼_{nilten/(N.O. Ilten).}

1 Tel.: +49 355693043; fax: +49 355693042.

0747-7171/$ – see front matter©2010 Elsevier Ltd. All rights reserved.

such codes. One class of varieties where non-trivial estimates have been made is that of toric varieties, which one can describe combinatorially.

Toric varieties have been generalized inAltmann and Hausen(2006) andAltmann et al.(2008) to so-calledT-varieties, which are normal varietiesXadmitting an effectivem-dimensional torus action. Form

=

dimXwe are in the case of toric varieties, but in generalmis supposed to be smaller than the dimension ofX.T-varieties can then be described by a varietyYof dimension dimX

−

malong with combinatorial data called a divisorial fan. If the acting torus has codimension one,Y is then a curve. The aim of this paper is to analyze certain evaluation codes on such varieties; we shall call these codes

T-codes.

In short, aT-code over_Fqis constructed from:

•

a curveYoverFq;

•

a so-calleddivisorial polytope(cf.Definition 15), essentially a concave functionh∗

:

_h

→

DivQY

wherehis a polytope with vertices in some latticeM

∼

=

Zmandh∗ satisfies some additional conditions;

•

and a setP

= {

P1

, . . . ,

Pl

}

ofFq-rational points onY.

Assuming that the support ofh∗

(

_u

)

_{is disjoint fromPfor eachu}

∈

h

∩

M, we can define theT-code C

(

Y

,

h∗

,

P

)

as the sum of a number of product codes:

C

(

Y

,

h∗

,

P

)

:=

X

u∈h∩M

Cu

⊗

C

(

Y

,

h∗

(

u

),

P

)

whereCuis the

[

(

q

−

1

)

m

,

1

, (

q

−

1

)

m

]

code generated by

(

tu

)

t∈(F∗q)m andC

(

Y

,

h

∗

(

u

),

P

)

is the AG code corresponding to the curveY, divisorh∗

(

_u

)

_{, and point setP. By interpretingC}

(

_Y

,

_h∗

,

_P

)

_{as the} image under a linear map of the Riemann–Roch space of a divisor on aT-variety, we are able to give non-trivial estimates for the dimensionkand minimum distancedof this code.

We begin in Section2by recalling the basic theory ofT-varieties. We then proceed to describe divisors and intersection theory onT-varieties in Section3. In particular, we describe allT-invariant Cartier and Weil divisors combinatorially, calculate the global sections of aT-invariant Cartier divisor, and determine exactly when aT-Cartier divisor is (semi-)ample. Furthermore, we provide formulae for calculating intersection numbers and for the Euler characteristic of a line bundle. The theory of this section is analogous to that of divisors on toric varieties and is essential for estimating the parameters of the evaluation codes we construct.

In Section4, we defineT-codes and show how to estimate the dimension and minimum distance, providing upper and lower bounds for both parameters. We give special attention to the case of two-dimensionalT-varieties, where we provide a better lower bound for the minimum distance.

Finally, we provide a number of examples in Section5. We first considerT-codes coming from those ruled surfaces corresponding to a rank-two decomposable vector bundle. In particular, we show that some of these codes have better parameters than those estimated for the product of a Reed–Solomon code and a one-point Goppa code. In a second example, we show how one can use the Hasse–Weil bound to improve the lower bound on the minimum distance. This example also shows that there are betterT-codes than those coming from ruled surfaces. In a final example, we describe aT-code over F7whose parameters are as good as any known linear code.

2. The theory ofT-varieties

First we recall some facts and notations from convex geometry. Here,Nalways is a lattice and

M

:=

Hom

(

N

,

_Z

)

its dual. The associated_Q-vector spacesN

⊗

_QandM

⊗

_Qare denoted byN_Qand

MQrespectively. Let

σ

⊂

NQbe a pointed convex polyhedral cone. A polyhedron∆which can be

written as a Minkowski sum∆

=

π

+

σ

of

σ

and a compact polyhedron

π

is said to have

σ

as its tail cone.

With respect to Minkowski addition the polyhedra with tail cone

σ

form a semigroup which we denote by Pol+_σ

(

N

)

. Note that

σ

∈

Pol+_σ

(

N

)

is the neutral element of this semigroup and that

∅

is by definition also an element of Pol+

A polyhedral divisor with tail cone

σ

on a normal varietyYis a formal finite sum

D

=

X

D

∆D

⊗

D

,

whereDruns over all prime divisors onY and∆D

∈

Pol+_σ. Here, finite means that only finitely many

coefficients differ from the tail cone.

We may evaluate a polyhedral divisor for every elementu

∈

σ

∨

∩

Mvia

D

(

u

)

:=

X

D

min v∈∆D

h

u

, v

i

D

in order to obtain an ordinary divisor on LocD. Here, LocD

:=

Y

\

S

∆D=∅D

denotes the locus ofD.

Definition 1. A polyhedral divisorDis calledCartierif every evaluationD

(

u

)

,u

∈

σ

∨

∩

M, is Cartier. To a Cartier polyhedral divisor we associate anM-gradedk-algebra sheaf and consequently an affine scheme over LocDadmitting aTM-action:

˜

X

:= ˜

X

(

D

)

:=

Spec_LocD

M

u∈_σ∨_∩M

O

(

D

(

u

)).

FromAltmann and Hausen(2006), we know that this construction gives a normal variety of dimension dimN

+

dimYadmitting a torus action ofTN_{with LocDas its good quotient.}

Moreover, for every affine normal varietyXthere exists a polyhedral divisorD such thatX

=

SpecΓ

(

X

˜

(

D

),

O_X(D)˜

)

.XandX

˜

coincide if LocDis affine. In this case LocDequals the categorical quotient ofX

˜

=

X. Definition 2. LetD

=

P

D∆D

⊗

D,D0

=

P

D∆ 0

D

⊗

Dbe two polyhedral divisors onY.

(1) We writeD0

⊂

Dif∆0_D

⊂

∆_Dholds for every prime divisorD. (2) We define the intersection of polyhedral divisors

D

∩

D0

:=

X

D

(

∆0

D

∩

∆D

)

⊗

D

.

(3) We define the degree of a polyhedral divisor degD

:=

X

D ∆D

.

(4) For a (not necessarily closed) pointy

∈

Ywe define the fibre polyhedron∆y

:=

Dy

:=

P

y∈D∆D.

(5) We callD0_{afaceofDand writeD}0

≺

_DifD0

yis a face ofDyfor everyy

∈

Y.

Assume thatD0

⊂

_D. This implies that

M

u∈σ∨_∩M

O

(

D0

(

u

))

←

-

M

u∈σ∨_∩M

O

(

D

(

u

)))

and we get a dominant morphismX

˜

(

D0

)

→ ˜

_X

(

_D

)

_.

Proposition 3 (Altmann et al.(2008), Proposition 3.4, Remark 3.5). This morphism defines an open em-bedding if and only ifD0

≺

Dholds.

Now we define the global analogue of a polyhedral divisor. The step from the affine to the complete case is reflected by the replacement of the polyhedra by complete polyhedral subdivisions. For every polyhedron in such a subdivision we get a corresponding tail cone. We will refer to the set of all tail cones as the tail fan of the subdivision.

(a)ΞQ1. (b)ΞQ2.

Fig. 1.The fansy divisor of a surface.

(a)Ξ0. (b)Ξ∞. (c)Ξ1.

Fig. 2.The fansy divisor of a threefold.

Definition 4. Consider a smooth projective curveY. Afansy divisoris a formal finite sum

Ξ

=

X

P∈Y ΞP

⊗

Z

such that:

(1) ΞPare polyhedral subdivisions coveringNQand sharing a common tail fan;

(2) Finite means here that for all but finitely many points,ΞPequals the tail fan.

Consider a finite set of polyhedral divisorsS, such thatD

_D0

∩

D

≺

_D0_{for every pair}

D

,

D0

∈

_S. Assume furthermore that their polyhedral coefficientsDPform the subdivisionsΞPof a fansy divisor.

From such a set we may construct a schemeX

˜

(

Ξ

)

by gluing X

(

D

)

s via

˜

X

(

D

)

← ˜

X

(

D

∩

D0

)

→ ˜

X

(

D0

).

Note that we had to check the cocycle condition; this is done inAltmann et al.(2008, Theorem 5.3). From Theorem 7.5 ibid. we know that we get a complete variety this way.

This variety is uniquely determined by the underlying fansy divisor. Different setsScorrespond to different open coverings. Therefore, we may denote the resulting variety byX

˜

(

Ξ

)

.

Theorem 5.6 inAltmann et al.(2008) tell us that for every normalT-varietyX with dimX

=

dimT

+

1 we may find a fansy divisorΞand a proper birational mapX

˜

(

Ξ

)

→

X. IfXhas categorical quotient of the expected dimension this morphism turns out to be the identity.

Remark 5. For a fansy divisorΞand an open covering

{

Ui

}

i∈IofYwe can find a setSas above, such

that for everyD

∈

Sthere is ai

∈

Isuch that LocD

=

Ui.

Example 6. LetY be a smooth projective curve andQ1

,

Q2

∈

Y two points. We consider the fansy divisorΞgiven by the coefficients inFig. 1.X

˜

(

Ξ

)

is a complete surface with one-dimensional torus action.

Example 7. We consider the fansy divisor on_P1_{given by the coefficients inFig. 2.X}

˜

(

_Ξ

)

_{is a complete}

(singular) threefold with two-dimensional torus action.

3. Divisors and intersection theory onT-varieties

From now on we shall only consider torus actions of codimension one; we will study them via fansy divisors.

3.1. Cartier divisors

LetΣ

⊂

NQbe a complete polyhedral subdivision ofNconsisting of tailed polyhedra. We consider

continuous functionsh

: |

Σ

| →

_Qwhich are affine on every polyhedron in Σ. Let ∆

∈

Σ

be a polyhedron with tail cone

δ

. Thenhinduces a linear functionh∆₀ on

δ

=

tail∆by defining

h∆₀

(v)

:=

h

(

P

+

v)

−

h

(

P

)

for someP

∈

∆. We callh∆₀ the linear part ofh

|

_∆.

Definition 8. An (integral) support function on a polyhedral subdivision Σ is a piecewise affine function as above with integer slope and integer translation. To be precise: for

v

∈ |

Σ

|

andk

∈

_N

such thatk

v

is a lattice point we havekh

(v)

∈

_Z. The group of support functions onΣis denoted by SF_Σ.

LetΞbe a fansy divisor onY. We consider SF

(

Ξ

)

, the group of formal sums

P

P∈YhPPwith the

following conditions.

(1) hP

∈

SFΞPa support function of theP-slice ofΞ. (2) allhPhave the same linear parth0.

(3) hPdiffers fromh0for only finitely many pointsP

∈

Y.

We refer to this fact by calling this sum finite and we omit those summands which equal h0.

Definition 9. A support functionh

∈

SF

(

Ξ

)

is called principal ifh

(v)

= h

u

, v

i +

D, withu

∈

MandD

is a principal divisor onY. Byh

(v)

= h

u

, v

i +

Dwe mean thathP

(v)

= h

u

, v

i +

aP, whereD

=

P

PaPP.

Ifh

=

P

hPP

∈

SF

(

Ξ

)

we consider a covering

{

Yi

}

ofYsuch thatPis a principal divisor on theYi

for everyP

∈

YwithhP

6=

h0, and such that everyYicontains at most one of these points.

We may find a setSas above which is compatible with this covering and inducesΞ. Now we choose aD

∈

Swith LocD

=

YiandhP

6=

h0.hPis an affine function on every polyhedron inΞP

so we get

−

hP

|

DP

(v)

= h

v,

u

i +

afor someu

∈

Manda

∈

Z. Assume that div

(

f

)

=

aPonYi; then

f

·

χ

u

∈

_K

(

_X

˜

(

_D

))

T_{defines aT-invariant principal divisorH}

D onX

˜

(

D

)

. These principal divisors fit together to a Cartier divisorDhonX

˜

(

Ξ

)

. HereK

(

X

˜

(

D

))

T

:=

L

u∈MK

(

Y

)

·

χ

u

⊃

Γ

(

X

˜

(

D

))

denotes the

ring of invariant rational functions onX

˜

(

D

)

. In this way the group of integral support functions onΞ corresponds to that of invariant Cartier divisors onX

˜

(

Ξ

)

.

3.2. Weil divisors

In general there are two types ofT-invariant prime divisors, namely (1) those which consist of orbit closures of dimension dimT; and (2) those which consist of orbit closures of dimension dimT

−

1.

Proposition 10. If D is a polyhedral divisor on a curve with tail cone

σ

, there are one-to-one correspondences

(1) between prime divisors of type1and pairs

(

P

, v)

with P a point on Y and

v

a vertex of∆P; and

(2) between prime divisors of type2and rays

ρ

of

σ

withdegD

∩

ρ

= ∅

.

Proof. Consider the quotient map

π

: ˜

X

→

LocD. InAltmann and Hausen(2006) the orbit structure of the fibres of

π

is described. Thus, we know that facesF

≺

Dycorrespond toT-invariant subvarieties

of codimension dim

(

F

)

in

π

y

:=

π

−1

(

y

)

. The correspondences follow by using this for closed points

and the generic point, respectively.

Remark 11. We may also describe the ideals of prime divisors in terms of polyhedral divisors: (1) For prime divisors of type1corresponding to a vertex

(

P

, v)

, the ideal is given by

IP,v

=

M

u∈_σ∨

(2) For prime divisors of type2, the corresponding ideal is generated by all multidegrees which are not orthogonal to

ρ

: I_ρ

=

M

u∈σ∨_\ρ⊥ Γ

(

Y

,

O

(

D

(

u

))).

Proposition 12. Let h

=

P

PhPcorrespond to the Cartier divisor DhonX

˜

(

D

)

. The corresponding Weil

divisor is given by

−

X

ρ h0

(

nρ

)ρ

−

X

(P,v)

µ(v)

hP

(v)(

P

, v),

where

µ(v)

is the smallest integer k

≥

1such that k

·

v

is a lattice point. This lattice point is a multiple of the primitive lattice vector n_v:

µ(v)v

=

ε(v)

n_v.

Proof. This is a local statement, so we will pass to a sufficiently small invariant open affine set which meets a particular prime divisor. If we translate this to our combinatorial language and we consider a prime divisor corresponding to

(

P

, v)

or

ρ

then we have to choose a polyhedral divisorD0

≺

_D

∈

_S such that

v

is also a vertex ofDP0or

ρ

is a ray in tailD

0

, respectively. So we restrict ourselves to the following two (affine) cases:

(1) Dis a polyhedral divisor with tail cone

σ

=

0 and a single point∆P

= {

v

} ⊂

Nas the only

nontrivial coefficient. Moreover,Yis affine and factorial. In particular,Pis a prime divisor with (local) parametertP.

(2) Dis the trivial polyhedral divisor with one-dimensional tail cone

ρ

over an affine locusY. In the first case we may chooseZ-basise1

, . . . ,

emofN withe1

=

nv. Consider the dual basis

e∗₁

, . . . ,

e∗_m. By definition

ε(v)

and

µ(v)

are coprime so we will finda

,

b

∈

_Zsuch thata

µ(v)

+

b

ε(v)

=

1. In this situationy

:=

ta P

χ

be ∗ 1 _{is irreducible in} Γ

(

OX

)

=

Γ

(

OY

)

[

y

,

t ±ε(v) P

χ

∓µ(v)e∗₁

, χ

±e∗₂

, . . . , χ

±e∗m

]

and defines the prime divisor

(

P

, v)

. We consider an elementt_Pα

χ

uwithu

=

P

i

λ

ie∗i. They-order of t_Pα

χ

u_is

ε(v)λ

1

+

µ(v)α

=

µ(v)(

h

u

, v

i +

α),

becauset_Pα

χ

u

=

yε(v)λ1+µ(v)α

(

_t−ε(v) P

χ

µ(v)e ∗ 1

)

λ1a+bα_{, and}

(

_t−ε(v) P

χ

µ(v)e ∗ 1

)

_{is a unit.}

In the second case we choose aZ-basise1

, . . . ,

emofNwithe1

=

nρ. We once again consider the dual basise∗₁

, . . . ,

e∗_m. In this situation

Γ

(

OX

)

=

Γ

(

OY

)

[

χ

e ∗ 1

, χ

±e ∗ 2

, . . . , χ

±e ∗ m

]

.

Now

(χ

e∗1

)

_{defines the prime divisor}

ρ

_{onX. For a principal divisorf}

·

χ

u, the

χ

e∗1_{-order equals the}

e∗

1-component ofu; i.e.,

h

u

,

nρ

i

.

Example 13. For our threefold example we considerDhwhereh0

,

h∞

,

h1are given by the tropical polynomials

h0

=

0

x(−1,0)

⊕

0

x(−1,1)

⊕

0

x(0,1)

⊕

0

x(1,0)

⊕

1

x(1,−1)

⊕

1

x(0,−1)

h∞

=

(

−

2

)

x(−1,0)

⊕

(

−

2

)

x(−1,1)

⊕

(

−

1

)

x(0,1)

⊕

(

−

1

)

x(1,0)

⊕

(

−

2

)

x(1,−1)

⊕

(

−

2

)

x(0,−1)

h1

=

1

x(−1,0)

⊕

1

x(−1,1)

⊕

0

x(0,1)

⊕

0

x(1,0)

⊕

0

x(1,−1)

⊕

0

x(0,−1)

where we are using the tropical semi-ring with operations

⊕ =

min

,

= +

. These support functions are pictured inFig. 3. The Weil divisor corresponding toDhis

P

ρDρ

+

2D₍∞,0)

+

2D(∞,(−1,−1)). This is the anti-canonical divisor ofX

:= ˜

X

(

Ξ

)

(Petersen and Süß,2008).

(a)h0. (b)h∞.

(c)h1.

Fig. 3.Support functions for aT-threefold.

3.3. Global sections

For a support functionhonXwe may consider theM-graded vector space of global sections ofDh

L

(

Dh

)

=

M

u∈M

L

(

Dh

)

u

:=

Γ

(

X

,

O

(

Dh

)).

Theweight setofL

(

Dh

)

is defined as the set

{

u

∈

M

|

L

(

D

)

u

6=

0

}

. For a Cartier divisor given by

h

∈

T-CaDiv

(

Ξ

)

we will bound its weight set by a polyhedron as well as describe the graded module structure ofL

(

D

)

.

Consider a support functionh

=

P

PhPPwith linear parth0. We define its associated polytope h

:=

h0

:= {

u

∈

MQ

| h

u

, v

i ≥

h0

(v)

∀

v∈N

}

and associate a dual functionh∗

:

h

→

DivQYvia

h∗

(

u

)

:=

X

P

h∗_P

(

u

)

P

:=

X

P

minvert

(

u

−

hP

)

P

,

where minvert

(

u

−

hP

)

denotes the minimal value ofu

−

hPon the vertices ofΞP.

Remark 14. Let hbe a concave support function. Every affine piece of hP corresponds to a pair

(

u

,

−

au

)

⊂

M

×

Z.h∗Pis defined to be the coarsest concave piecewise affine function withh

∗

P

(

u

)

=

au.

We can reformulate this in terms of the tropical semi-ring with operation

⊕ =

min

,

= +

. We might think of thehPas given by tropical polynomials

L

w∈I

(

−

aw

)

xw; thenh

=

conv

(

I

)

and

h∗

P

(w)

=

aw, i.e.,Γh∗_P is the reflected lower Newton boundary of the tropical polynomial forhP.

Definition 15. Adivisorial polytope h∗_{is a pair consisting of an ordinary polytope}

h

⊂

MQand a

concave piecewise affine functionh∗

:

h

→

DivQYsuch that

(1) degh∗

(

_u

)

≥

_{0 for all verticesuof} h, and

(2) some multiple ofh∗

(

_u

)

_{is principal in the case of degh}∗

(

_u

)

=

_{0 for a vertexu.} (3) his a lattice polytope as is conv

(

Γh∗

(a)h∗ 0. (b)h ∗ ∞. (c)h ∗ 1. Fig. 4.h∗ for aT-threefold. (a)Ξ0 0. (b)Ξ 0 ∞. (c)Ξ 0 1. Fig. 5.A refined polyhedral divisor.

Letg

,

h

∈

MQbe polytopes. For any concave piecewise affine functionsg

∗

:

g

→

DivQY and

h∗

:

h

→

DivQYwe define their sumg

∗

+

h∗to be the piecewise affine concave function ong

+

h

given by

(

g_P∗

+

h∗_P

)(

u

)

=

max

{

h∗_P

(w)

+

g_P∗

(w

0

)

|

u

=

w

+

w

0

}

.

Remark 16. Forg

,

h

∈

SF

(

Ξ

)

, one easily checks that

g

+

h

⊂

g+h

and that

g_P∗

(

u

)

+

h∗_P

(

u

)

≤

(

g

+

h

)

∗_P

(

u

)

for allP

∈

Yand allu

∈

_g

+

_h. Furthermore, ifhPandgPare convex, they correspond to tropical

polynomials f,f0_{. It follows then that}

(

_g

+

_h

)

P corresponds to f

f0. Its reflected lower Newton

boundary is exactly the graph of

(

g

+

h

)

∗_P; thus the equality

(

g

+

h

)

∗_P

=

g_P∗

+

h∗_P

holds.

To a divisorial polytopeh∗ we might associate a fansy divisorΞ and support functionhonΞ such thath∗_{corresponds tohin the way given above. Indeed, to everyh}∗

Pwe can associate a tropical

polynomial f

:=

L

(u,au)

(

−

au

)

x

u_{, where}

(

_u

,

_a

u

)

runs over the vertices ofΓ(h∗_P). This polynomial

induces via evaluation a piecewise affine function and a polyhedral subdivisionΞPofN.

Remark 17. If we remove condition3from the definition of a divisorial polytope (Definition 15), the association in the above paragraph gives us aQ-Cartier divisor.

For every fansy divisor there exists a smooth refinement, i.e. a fansy divisorΞ0such that everyΞ_P0 is a refinement ofΞPandX

˜

(

Ξ0

)

is smooth (Süß,2008). Every support functionhonΞis obviously also

a support function onΞ0_{. Thus, for a given divisorial polytopeh}∗_{we might always consider a smooth} fansy divisorΞand a support functionhon it such that the associated dual function equalsh∗_. Example 18. We now revisit our threefold example.Fig. 4shows a sketch ofh∗. We show a refinement of the fansy divisor inFig. 5which gives a smooth threefold.

Proposition 19. Let h

∈

SF

(

Ξ

)

be a Cartier divisor with linear part h0. Then (1) the weight set of L

(

Dh

)

is a subset ofh; and

(2) for u

∈

hwe have

Proof. By the definition ofO

(

Dh

)

we have Γ

(

X

,

O

(

Dh

))

T

=

(

χ

u_f

|

_div

(χ

u_f

)

−

X

ρ h0

(

nρ

)ρ

−

X

(P,v)

µ(v)

hP

(v)(

P

, v)

≥

0

)

.

But div

(χ

uf

)

=

P

ρ

h

u

,

nρ

i

ρ

+

P

(P,v)

µ(v)(

h

u

, v

i +

ordP

(

f

))(

P

, v)

, so for

χ

uf

∈

L

(

h

)

we get the

following bounds: (1)

h

u

,

n_ρ

i ≥

h0

(

nρ

)

∀

ρ

(2) ordP

(

f

)

+ h

u

, v

i ≥

hP

(v)

∀

(P,v).

The first implies thatu

∈

h

∩

M, and the second that ordP

(

f

)

+

(

u

−

hP

)(v)

≥

0

∀

(

P

, v)

.

For a cone

σ

∈

Ξ₀(n)of maximal dimension in the tail fan and aP

∈

Ywe get exactly one polyhedron

∆σ

P

∈

ΞPhaving tail

σ

. For a given concave support functionh

=

P

hPP, we have

hP

|

∆σP

= h·

,

u

h

(σ )

i +

_ah P

(σ ).

The constant part gives rise to a divisor onY:

h

|

_σ

(

0

)

:=

X

P

ah_P

(σ)

P

.

Proposition 20. A T -Cartier divisor h

=

P

hPP

∈

T-CaDiv

(

Ξ

)

is (semi-)ample if and only if all hP

are strictly concave (concave) and

−

h

|

_σ

(

0

)

is (semi-)ample for all tail cones

σ

, i.e.,deg

−

h

|

_σ

(

0

)

=

−

P

Pa h

P

(σ) >

0(or a multiple of

−

h

|

σ

(

0

)

is principal).

Proof. We first prove that semi-ampleness follows from the above criteria. Becausehis (strictly) concave the same is true forh0. This implies that theuh

(σ)

are exactly the vertices of h and

h∗

(

uh

(σ))

=

h

|

_σ

(

0

)

.

The semi-ampleness forh∗

(

_u

),

_u

∈

h

∩

M follows from the semi-ampleness at the vertices.

Indeed ifD

,

D0_{are semi-ample divisors onYthis is also true forD}

+

λ(

_D0

−

_D

)

_{with 0}

≤

λ

≤

_1. Every vertex

(

u

,

au

)

ofΓh∗_Pcorresponds to an affine piece ofhPof the form

h

u

,

·i −

au. If we letfbe

such that div

(

f

)

=

a_uPon LocDfor someD

∈

_S, we then haveD_h

|

_X˜_(D)

=

div

(

f−1

χ

−u

)

(see3.1). A point

(

u

,

au

)

∈

M

×

Zis a vertex ofh∗exactly if

(

ku

,

kau

)

is a vertex of

(

k

·

h

)

∗. Hence, after passing to a

suitable multiple ofhwe may assume, thath∗

(

_u

)

_{is base-point free withfbeing a global section which} generatesO

(

h∗

(

u

))

on LocD. Thusf

χ

uis a global section ofO

(

Dh

)

which generatesO

(

Dh

)

|

_X˜_(D).

To show the other direction, i.e. that semi-ampleness implies the above criteria, assume thathPis

not concave. Then this is true also for every multiple of

`

·

hPand hence there is an affine piece

h

u

,

·i−

au

of

`

hPsuch thatau

> (`

hP

)

∗

(

u

)

. This means there is no global sectionf

χ

usuch that div

(

f

)

=

auP. But

this contradicts the base-point freeness ofD_`hand hence the semi-ampleness ofDh.

To get the statement for ampleness note that a support functionhon a polyhedral subdivision is strictly concave if and only if for every support functionh0there is ak

0 such thath0

+

khis concave.

Corollary 21. X

˜

(

Ξ

)

is projective if and only if allΞPare regular subdivisions, i.e. admit a strictly convex

support function.

Remark 22. We see fromProposition 20that forh

∈

SF

(

Ξ

)

, if theT-invariant divisorDhis

semi-ample, the corresponding dual functionh∗_{is in fact a divisorial polytope. Conversely, ifh}∗_{is a divisorial} polytope, the associated divisor on the associatedT-variety is semi-ample.

3.4. Intersection numbers

Definition 23. For a divisorial polytopeh∗we define itsvolumeto be

volh∗

:=

X

P

Z

h

For divisorial polytopesh∗ 1

, . . . ,

h

∗

kwe define theirmixed volumeby

V

(

h∗₁

, . . . ,

h∗_k

)

:=

k

X

i=1

(

−

1

)

i−1

X

1≤j1≤···ji≤k vol

(

h∗_j1

+ · · · +

h∗_j i

).

Proposition 24. Assume that on X Kodaira’s vanishing Theorem holds.

(1) If Dhis semi-ample, for the self-intersection number we get

(

Dh

)

(m+1)

=

(

m

+

1

)

!

volh∗

.

(2) Let h1

, . . . ,

hm+1define semi-ample divisors Dion X

(

Ξ

)

. Then

(

D1

· · ·

Dm+1

)

=

(

m

+

1

)

!

V

(

h∗1

, . . . ,

h ∗

m+1

).

Proof. If we apply (1) to every sum of divisors fromD1

, . . . ,

Dm+1we get (2) by the multi-linearity and symmetry of intersection numbers.

To prove (1) we first recall that

(

Dh

)

m+1

=

lim

ν→∞

(

m

+

1

)

!

ν

m+1

χ(

X

,

O

(ν

Dh

)),

but for projective X

:=

X

(

Ξ

)

and nef divisors the ranks of higher cohomology groups are asymptotically irrelevantDemailly(2001, Theorem 6.7) so we get

(

Dh

)

m+1

=

_νlim →∞

(

m

+

1

)

!

ν

m+1 h 0

(

_X

,

_O

(ν

_D h

)).

Note that

(ν

h

)

∗

(

_u

)

=

ν

·

_h∗

(

1

νu

)

. Now we can boundh0by

X

u∈νh∩M deg

b

ν

h∗ 1_νu

c −

g

(

Y

)

+

1

≤

h0

(

O

(ν

Dh

))

≤

X

u∈νh∩M deg

b

ν

h∗ 1_νu

c +

1

.

(1) On the one hand we have

lim ν→∞

(

m

+

1

)

!

ν

m+1

X

u∈νh∩M deg

b

ν

h∗ _ν1u

c =

lim ν→∞

(

m

+

1

)

!

ν

m

X

u∈h∩1_νM 1

ν

deg

b

ν

h ∗

(

u

)

c

=

(

m

+

1

)

!

Z

h h∗volMR

.

On the other hand, for any constantc, we have

lim ν→∞ 1

ν

m+1

X

u∈νh∩M c

=

c

·

lim ν→∞ #

(ν

·

h

∩

M

)

ν

m+1

=

0

.

Thus, if we pass to the limit in (1), the term in the middle has to converge to volh∗_.

Remark 25. The theorem allows us to compute intersection numbers in characteristic 0 as well as onT-surfaces in positive characteristic because Kodaira’s vanishing theorem holds in these cases. We believe that the theorem holds as well for positive characteristic in higher dimensions; work is being done to show that the vanishing theorem holds there.

Corollary 26. Let h

∈

SF

(

Ξ

)

and let C be any one-cycle rationally equivalent to the intersection of Cartier divisors, each of which can be expressed as an integer linear combination of semi-ample Cartier divisors. Then Dh

·

C is equal to Dh+P−Q

·

C for all points P

,

Q

∈

Y .

Proof. We have

Dh+P−Q

·

C

=

(

Dh

−

D−P

+

D−Q

)

·

C

=

Dh

·

C

−

D−P

·

C

+

D−Q

·

C

so it is sufficient to show thatD−P

·

C

=

D−Q

·

C. Now,D−PandD−Qare semi-ample, so we can apply

Proposition 24. Using the fact that vol

((

−

P

)

∗

+

e

h∗

)

=

vol

((

−

Q

)

∗

+

e

h∗

)

for all

e

h

∈

SF

(

Ξ

)

gives the

desired equality.

Example 27. We know byProposition 20thatDhin our threefold is ample. We have volh∗

=

21.

3.5. Genus of curves on surfaces

LetX

= ˜

X

(

Ξ

)

be a two-dimensionalT-variety and leth

∈

SF

(

Ξ

)

be a support function onΞ. For any curveC

∈ |

Dh

|

, we show how to calculate the arithmetic genusg

(

C

)

. As a corollary, we can

calculate the Euler characteristic

χ(

X

,

O

(

Dh

))

ifXis smooth.

Definition 28. For anyh

∈

SF

(

Ξ

)

, let inth∗_P

:=

X

u∈◦h∩M #

{

a

∈

_Z≥0

|

a

<

|

h∗_P

(

u

)

|} ·

h ∗ P

(

u

)

|

h∗_P

(

u

)

|

for each pointP

∈

Y, where◦_his the interior ofh. Furthermore, let

inth∗

:=

X

P∈Y

inth∗_P

.

Definition 29. For anyh

∈

SF

(

Ξ

)

, let #h∗_P

:=

X

u∈h∩M

b

h∗_P

(

u

)

c

for any pointP

∈

Yand let #h∗

:=

X

u∈h∩M

deg

b

h∗

(

u

)

c =

X

Y∈P

#h∗_P

.

Remark 30. Note that inth∗

Pis the number of ‘‘interior’’ lattice points between the graph ofh

∗

Pand 0

counted with their signs, where lattice points in height 0 are counted as long as they are not on the boundary ofh. Similarly, if #h∗P

(

h

)

≥

0 for allu

∈

h, #h∗Pis the sum of the number of lattice points

between the graph of #h∗

Pand 0, where we count no lattice points in height 0 but all lattice points

lying on the graph ofh∗_P.

We will use the following lemma.

Lemma 31. With notation as above,2

·

volh∗_P

=

inth∗_P

+

#h∗_Pfor all P

∈

Y . It follows in particular that

2

·

volh∗

=

_inth∗

+

_#h∗_.

Proof. Fix someP

∈

Y. Suppose now thath∗_P

(

u

)

≥

0 for allu

∈

hand set

∆

=

conv

(

u

,

h∗_P

(

u

))

∪ {

(

u

,

0

)

}

,

whereu

∈

h. This is a convex polytope inMQ0, whereM

0

=

M

×

_Z. Pick’s theorem tells us that 2

·

vol∆

+

2

=

#

(

∆

∩

M0

)

+

_#

(

_∆◦

∩

_M0

)

_{. Now vol∆}

=

_volh∗

P, #

(

∆

∩

M

)

=

#h

∗

P

+

#

(

h

∩

M

)

,

and #

(

∆◦

∩

M

)

=

inth∗_P

−

#

(

h

∩

M

)

+

2, so the desired equality follows. For generalh∗P, choosej

such that

e

h∗_P

(

u

)

:=

h∗_P

(

u

)

+

j

≥

0 for allu

∈

h. Then 2

·

vol

e

h∗_P

=

int

e

h∗_P

+

#

e

h∗_Pand forj∗_P

(

u

)

:=

jwe

have 2

·

volj∗_P

=

intj∗_P

+

#j∗_P. Since vol, int, and # are additive at least for integer-valued functions, the desired equality follows forh∗_P

=

e

h

∗

P

−

j

∗

P.

We are now able to prove the following proposition:

Proposition 32. Let h

∈

SF

(

Ξ

)

be any support function such that Dhis semi-ample. Then for C

∈ |

Dh

|

,

the arithmetic genus of C is given by

g

(

C

)

=

inth∗

+

1

+

volh

·

(

g

(

Y

)

−

1

),

where g

(

Y

)

is the genus of Y .

Proof. Without loss of generality, we can take the curveCto equalDh. Indeed, the arithmetic genus

is invariant under rational equivalence and since

|

Dh

|

is not empty, it must contain someT-invariant

effective divisor. We compare the genus ofCwith that of a comparable curveC0onX0

:=

Y

×

P1and then compute the genus ofC0directly. To begin with, note that we can find monoidal transformations

π

i

:

Xi

→

Xi−11

≤

i

≤

ksuch that

(1) Xiis aT-variety;

(2)

π

iisT-equivariant; and

(3) there is a birationalT-equivariant morphism

ϕ

:

Xk

→

X.

This is done as follows. LetΣbe the fan

{

_Q≥0

,

_Q≤0

,

{

0

}}

and letΞ0

P

:=

Σfor all pointsP

∈

Y. Then

X0

= ˜

X

(

Ξ0

)

. Each morphism

π

icorresponds to an additional subdivision in the fanΞi−1at exactly

one point. Thus, we keep on refining until we get aΞkwhich is a smooth common refinement ofΞ andΞ0_{; this gives us our morphism}

ϕ

_{. Finally, we let}

π

:

_X

k

→

X0be the composition of the

π

i.

We now pull backCtoCk

:=

ϕ

∗

(

C

)

. Thus we now haveCk

=

Dh, wherehis now considered as a

support function onΞk. Furthermore, this does not change the arithmetic genus; that is,g

(

C

)

=

g

(

Ck

)

.

Define now inductivelyCi−1

=

π

i∗

(

Ci

)

for 1

≤

i

≤

k. One easily checks thatC0

=

Deh, where

e

h

∈

SF

(

Ξ0

)

is the support function given by the divisorial polytope

e

h∗_P

:=

maxu∈hh

∗

P

(

h

)

with

eh

:=

h. Note that sinceCis semi-ample, eachCiis semi-ample as well. We will now calculate

the difference betweeng

(

Ck

)

andg

(

C0

)

.

We first consider a special case; namely, suppose thath∗

P is trivial everywhere except for at two

pointsQ1

6=

Q2. IfY

=

P1, all the varietiesXi andXare toric. In this case, the divisorDhcan be

understood in toric terms as the polytope

∆h

:=

convΓh∗

Q₁

∪

Γ−h∗Q₂

andDehcorresponds to∆eh, which is defined in a similar manner. Then

g

(

Ck

)

−

g

(

C0

)

=

I

(

∆h

)

−

I

(

∆eh

),

whereI

(

∆

)

is the number of interior lattice points of∆; see for exampleLittle and Schenck(2006), prop. 5.1. But we haveI

(

∆h

)

=

inth∗Q1

+

inth

∗

Q2

−

#

(

◦

h

∩

M

)

and a similar equation for

e

h, which leads

to

g

(

Ck

)

−

g

(

C0

)

=

inth∗

−

int

e

h ∗

.

(2) Now, Eq. (2) actually holds in general, not just in the toric case. To see this, note that for each 1

≤

i

≤

k,Ci

=

π

i∗

(

Ci−1

)

+

ri

·

Ei, whereEiis the exceptional divisor of

π

i. Then similar toHartshorne

(1977), V.3.7 we haveg

(

C_i

)

=

g

(

C_i−1

)

−

12ri

(

ri

+

1

)

. Thus, g

(

Ck

)

−

g

(

C0

)

=

k

X

i=1

−

1 2ri

(

ri

+

1

).

However, for each 1

≤

i

≤

k, the integerrican be determined combinatorially by comparing the

polyhedral subdivisionsΞ_Pi andΞ_Pi−1for the single pointP

∈

Y where these fansy divisors differ. Thus, the integersrican be calculated exactly as if we were in the toric case, so we get

k

X

i=1

−

1 2ri

(

ri

+

1

)

=

inth ∗

−

int

e

h∗

.

Eq. (2) follows.

We now calculateg

(

C0

)

. From the adjunction formula, we have

g

(

C0

)

=

D2

eh

+

Deh

·

K0

2

+

1

forK0a canonical divisor onX0; seeHartshorne(1977, V.1.5). The theorem of Riemann–Roch for surfaces (Hartshorne,1977, V.1.6) gives us

χ(

X0

,

O

(

Deh

))

=

D2 eh

−

Deh

·

K0 2

+

χ(

X0

,

OX0

).

Thus, g

(

C0

)

=

De2h

+

1

+

χ(

X0

,

OX0

)

−

χ(

X0

,

O

(

Deh

)).

Now,

χ(

X0

,

OX0

)

=

1

−

g

(

Y

)

(seeHartshorne(1977), V.2.5). Likewise, ifp

:

X0

→

Yis the projection, we have

χ

X0

,

O

(

Deh

)

=

χ

Y

,

p∗O

(

Deh

)

=

X

u∈h∩M

χ(

Y

,

O

(

e

h∗

(

u

)))

=

#

e

h

+

(

1

−

g

)

·

(

vol_h

+

1

),

where the last equation follows from the Riemann–Roch theorem for curves. We also haveD2

eh

=

2

·

vol

e

h. Making these substitutions results in

g

(

C0

)

=

2

·

vol

e

h

+

1

+

vol_h

·

(

g

(

Y

)

−

1

)

−

#

e

h

=

int

e

h

+

1

+

volh

·

(

g

(

Y

)

−

1

),

the second equality coming fromLemma 31. Combining this with Eq. (2) completes the proof. Corollary 33. For any semi-ample T -invariant Cartier divisor Dhon a smooth T -variety X , we have

χ(

X

,

O

(

Dh

))

=

#h∗

−

(

g

(

Y

)

−

1

)

·

#

(

h

∩

M

)

=

X

u∈h∩M

χ(

Y

,

O

(

h∗

(

u

))).

Proof. Using the adjunction formula and the Riemann–Roch theorem for surfaces as in the above theorem gives us the formula

χ (

X

,

O

(

Dh

))

=

D2h

+

1

+

χ(

X

,

OX

)

−

g

(

C

)

for someC

∈ |

Dh

|

. We can use the above proposition to calculateg

(

C

)

. Combining this with the facts

thatD2

h

=

2

·

volhand

χ(

X

,

OX

)

=

1

−

g

(

Y

)

along withLemma 31completes the proof of the first

equality. The second equality follows directly from the theorem of Riemann–Roch for curves. At the end of this section we revisit our surface example and use it to illustrate the concepts we have introduced.

Example 34.We look at the Cartier divisorDhon our surface example wherehQ1andhQ2are given by

the tropical polynomials 0

⊕

(

−

2

)

x4_{and 0}

⊕

(

−

₂

)

_x2

⊕

(

−

₁

)

_x3

⊕

₁

_x4_{, respectively. One easily} sees thath

= [

0

,

4

]

, and thath∗Q1andh

∗

Q1 respectively correspond to the tropical polynomials x

1/2 and x

⊕

4

x−1

⊕

₇

_x−2_{. In other words,h}∗

Q1

(

u

)

=

u

/

2 and

h∗_Q2

(

u

)

=

(

_{u ifu}

≤

₂

4

−

u if 2

≤

u

≤

3 7

−

2u ifu

≥

3

.

InFig. 6we sketchhand the corresponding divisorial polytopeh∗_.

We can useProposition 12to compute the corresponding Weil divisor: 4D_Q≤0

+

4D₍Q2,2)

+

7D(Q2,1).

Dhis semi-ample, so byProposition 24we get

(

Dh

)

2

=

15. Finally, fromProposition 32we know that

a section ofDhhas genus 5

+

4

·

g

(

Y

)

.

We may also start withh∗_{and take the dualhto construct a fansy divisor as described above. We} recoverΞthis way.X

:= ˜

X

(

Ξ

)

is not smooth, but a refinement of the polyhedral subdivisions (see Fig. 7) gives a smooth surfaceX0_{(this is will not be proved here; seeSüß(2008)). UsingCorollary 33,} we can calculate that

χ(

X0

,

_O

(

_D

h

))

=

12

−

5

·

g

(

Y

)

.

4. T-codes and their parameters

4.1. Construction

LetYbe a curve overFqand leth∗be a divisorial polytope. LetP

= {

P1

, . . . ,

Pl

}

be some subset

of the_Fq-rational points ofYsuch that fori

=

1

, . . . ,

l,h∗Piis affine andh

∗

(a)hQ1. (b)hQ2. (c)h∗ Q1. (d)h ∗ Q2. Fig. 6.handh∗ for aT-surface. (a)Ξ0 Q1. (b)Ξ 0 Q2.

Fig. 7.A refined fansy divisor.

LetΞbe the fansy divisor associated toh∗and letΞ0be some minimal refinement such thatX

:=

˜

X

(

Ξ0

)

_{is smooth. Note that for each pointP}

i

∈

P,ΞP0i

=

v(

Pi

)

+

Σ, for a unique lattice point

v(

Pi

)

and tail fanΣ. Setm

=

dimM. For each pointPi, letPi1

, . . . ,

P

(q−1)m

i be the

(

q

−

1

)

m

Fq-rational points on

Xof the openT-orbit contracting toPi.

The support functionhassociated toh∗_{corresponds to a semi-ampleT-invariant}

Fq-rational Cartier

divisorDhonX. We denote the corresponding line bundle byO

(

Dh

)

and letL

(

Dh

)

=

Γ

(

X

,

O

(

Dh

))

. For

each pointP_ij, fix some isomorphismO

(

Dh

)

_Pj i

∼

=

_Fq. Consider theFq-linear map

ev

:

L

(

Dh

)

→

Fl(q−1) m q f

7→

f_P1 1

,

fP 2 1

, . . . ,

fP(q−1) m l

,

wheref

P_ijis the image off inFqfollowing the identification withO

(

Dh

)

P_ij. In other words, the above

map evaluates the rational functionf at thel

(

q

−

1

)

m_pointsPj

i 1

≤

i

≤

l, 1

≤

j

≤

(

q

−

1

)

m. The

image of ev is a linear subspace ofFl(q−1) m

q and thus a linear code of lengthn

=

l

(

q

−

1

)

m; we denote

it byC

(

Y

,

h∗

,

P

)

. IfP is maximal, we simply denote it byC

(

Y

,

h∗

)

_{. Note that although}

C

(

Y

,

h∗

,

P

)

indeed depends on the way we identifyO

(

Dh

)

_Pj i

distanceddo not. Thus, we will always assume that some such isomorphisms are given, but will not concern ourselves further with them.

Remark 35. Ifh∗

Pi

=

0 fori

=

1

, . . . ,

l, thenC

(

Y

,

h

∗

,

_P

)

_{is equivalent as code to the image of the map} ev

:

M

u∈h∩M Γ O

(

h∗

(

u

))

χ

u

→

Fl(q−1) m q g

χ

u

7→

g

(

P1

)χ

u

(

Q1

),

g

(

P1

)χ

u

(

Q2

), . . . ,

g

(

Pl

)χ

u

(

Q(q−1)m

)

whereQ1

, . . . ,

Q(q−1)m are the_Fq-rational points of them-dimensional torus. Thus, in this case the isomorphismsO

(

Dh

)

_Pj

i

∼

=

_Fqare not only irrelevant but also unnecessary. Now letCube the

[

(

q

−

1

)

m

,

1

, (

q

−

1

)

m

]

code generated by

(

tu

)

_t∈(F∗q)mand letC

(

Y

,

h

∗

(

u

),

P

)

be the AG code corresponding to the curveY, divisorh∗

(

_u

)

_{, and point setP. Then as mentioned in the introduction, we can also} defineC

(

Y

,

h∗

,

P

)

simply as

C

(

Y

,

h∗

,

P

)

=

X

u∈h∩M

Cu

⊗

C

(

Y

,

h∗

(

u

),

P

).

4.2. Estimate on dimension

Assume that the map ev is injective. This is always the case if the bound given below for the minimum distance is larger than zero. We then have

k

=

dim_FqL

(

Dh

).

UsingProposition 19, we thus get

k

=

X

u∈h∩M

dimΓ

(

Y

,

O

(

h∗

(

u

))).

We can approximatekusing only the combinatorics ofh∗. Let

γ (

u

)

=

(

_deg

b

_h∗

(

_u

)

c +

₁

−

_g

(

_Y

)

_{if deg}

b

_h∗

(

_u

)

c +

₁

−

_g

(

_Y

) >

₀

1 if deg

b

h∗

(

_u

)

c +

₁

−

_g

(

_Y

)

≤

_{0 andh}∗

(

_u

)

≥

₀ 0 if otherwise.

Proposition 36. If the evaluation mapevis injective, then

#h∗

+

#

(

h

∩

M

)(

1

−

g

)

≤

X

u∈h∩M

γ (

u

)

≤

k

≤

#h∗

+

#

(

h

∩

M

).

(3) Furthermore, k

=

#h∗

+

#

(

h

∩

M

)(

1

−

g

))

(4)

ifdegh∗

(

_u

) >

_2g

(

_Y

)

−

_{2for all u}

∈

h

∩

M.

Proof. The leftmost inequality in (3) follows from the definition of

γ (

u

)

. We now consider the second inequality in (3). Fix some degreeu

∈

h

∩

M. Then we always have dimΓ

(

Y

,

O

(

h∗

(

u

)))

≥

0, and

ifh∗

(

u

)

is effective, then dimΓ

(

Y

,

O

(

h∗

(

u

)))

≥

1. Using the theorem of Riemann and Roch (see for exampleHartshorne(1977)), we also have dimΓ

(

Y

,

O

(

h∗

(

u

)))

≥

degh∗

(

u

)

+

1

−

g, and the inequality follows. If degh∗

(

_u

) >

_2g

(

_Y

)

−

_{2, then equality holds, so (4) follows. Finally, the right inequality in} (3) follows from dimΓ

(

Y

,

O

(

h∗

(

_u

)))

≤

_degh∗

(

_u

)

+

_1.

4.3. General lower bound on minimum distance

One strategy to produce an estimate fordis to use techniques of intersection theory, as first presented inHansen(2001). These techniques have been applied to toric varieties; see for example

Hansen(2002) andRuano(2007). We first consider the general case and then specialize to surfaces.2 Lete∗₁

, . . . ,

e∗_mbe a basis forM. ForP

∈

Pand

η

1

, . . . , η

m−1

∈

F∗q, definel

(

q

−

1

)

m−1curves

CP,η1,...,ηm−1

:=

(

P

, v(

P

))

∩

V

{

χ

e∗i

−

η

_i

}

m_i=1−1

.

Each pointP_ijlies on exactly one of these curves. Furthermore, each curveCP,η1,...,ηm−1 is rationally

equivalent to CP

:=

(

P

, v(

P

))

∩

V

{

χ

e∗i

}

m_i=1−1

=

D0−P

·

(

D−e∗1

)

≥0

·

. . .

·

(

D−e∗m−1

)

_≥0

where the second equality follows fromProposition 12,e∗_i is considered as an element ofSF

(

Ξ

)

, and

(

D−e∗i

)

≥0is the effective part ofD−e∗i.

Fix some sections

∈

L

(

Dh

)

; this corresponds to an effective divisor

(

s

)

0

=

Dh

+

(

s

)

. ByZ

(

s

)

we denote the number of pointsP_ijsuch thats_Pj i

=

0. Equivalently,Z

(

s

)

is the number of pointsP_ij

contained in the support of

(

s

)

0. Thus, one has the following lower bound for the minimum distance:

d

≥

l

(

q

−

1

)

m

−

max

s∈L(Dh)

Z

(

s

).

Let

(

s

)

0 vanish on exactly

λ

of the curves

{

CP,η1,...,ηm−1

}

. Following (Hansen,2001) and setting

C

=

CPfor someP

∈

Pwe then have

Z

(

s

)

≤

λ(

q

−

1

)

+

(

l

−

λ)

Dh

·

C (5)

since

(

s

)

0

∼

Dhand it follows fromCorollary 26thatDh

·

C

=

Dh

·

CPi

=

Dh

·

CPi,η1,...,ηm−1for all 1

≤

i

≤

l.

Assuming that Kodaira’s vanishing theorem holds onX, we can useProposition 24to calculateDh

·

C.

We now bound

λ

in a method similar toRuano(2007). For the divisorial polytopeh∗

:

h

→

DivQY

let pr

(

h

)

be the projection ofhtoM

/

Ze∗mand define pr

(

h

∗

)

:

pr

(

h

)

→

DivQ

(

Y

)

by

pr

(

h∗

)

P

(

u

)

=

max

(u,um)∈h∩M

h∗_P

((

u

,

um

)).

One easily checks that pr

(

h∗

)

is a divisorial polytope. Assume thath

⊂

e

u

+ {

u

∈

M

|

0

≤

ui

≤

q

−

2

}

for some

e

u

=

(

e

u1

, . . . ,

e

um

)

∈

M. This also then holds for pr

(

h

)

. We can write

s

=

χ

eume ∗ m

·

s0

+

s1

χ

e ∗ m

+

_s q−2

χ

(q−2)e ∗ m

wheresi

∈

K

(

Y

)(χ

u1

, . . . , χ

um−1

)

. In fact, one easily checks thatsi

∈

L

(

Dpr(h)

)

, whereDpr(h)is the

T-invariant Cartier divisor on them-dimensionalT-varietyX_pr(h∗₎overYboth determined by pr

(

h∗

)

.

If we restricts

·

χ

−eume ∗ m_{to some curveC} P,η1,...,ηm−1we get a polynomials

=

s0

+

s1

χ

e∗m

+

_s q−2

χ

(q−2)e ∗ m

∈

Fq

[

χ

em

]

of degree less than or equal toq

−

2. IfCP,η1,...,ηm−1is a curve wheresvanishes, thenshasq

−

1

zeros, sos

≡

0 andsi

=

0 for 0

≤

i

≤

q

−

2. Thus the sectionsi

∈

L

(

Dpr(h)

)

vanishes on the point of

X_pr(h∗₎corresponding to the tuple

(

P

, η

₁

, . . . , η

_m−1

)

. It follows that

λ

≤

max

t∈L(Dpr₍h))

Z

(

t

).

Thus, we can recursively bound

λ

until dim

(

X

)

=

2.

4.4. Lower bound on minimum distance fordim

(

X

)

=

2

We can provide a much better bound forZ

(

s

)

whenXis a surface. Consider a global sectionsof

O

(

Dh

)

as before such that

(

s

)

0vanishes on exactly

λ

of the curves

{

CPi

}

, sayCQ1

, . . . ,

CQλwhere theQi

2 A more recent strategy to estimatedfor toric surface codes involves bounding the number of irreducible components of a section and then applying the Hasse–Weil bound, see for exampleLittle and Schenck(2006) andSoprunov and Soprunova

are distinct points inP. Thus,s

∈

L

(

Deh

)

, where

e

h

=

h

+

P

λ i=1Qi. Since

e

hand

P

λ i=1

(

−

Qi

)

are concave, it follows thath∗

=

e

h∗

+

(

P

λ

i=1

(

−

Qi

))

∗. In particular, we have that

deg

e

h ∗

(

u

)

=

degh∗

(

u

)

−

λ.

Thus,scan only have support in the weightsu

∈

(h,λ), where

(h,λ)

=

u

∈

h

∩

M

|

deg

b

h∗

(

u

)

c ≥

λ

.

It follows immediately that

λ

≤

max

u∈h∩M

deg

b

h∗

(

u

)

c :=

λ

₀

.

Having found a good bound for

λ

, we now try to improve on the upper bound forZ

(

s

)

in equation Eq. (5). By choosing a generator we can identify the latticeNwithZ. Then

σ

−

:=

Q≤0and

σ

+

:=

Q≥0 are the two rays inΣ. Each of these rays corresponds to a T-invariant divisor. Let

µ

− and

µ

+ respectively be the coefficients of the prime divisors

σ

−and

σ

+in

(

s

)

0. We want to find a lower bound for the sum

µ

−

+

µ

+. This is easy ifshas support only in a single weightu, says

=

f

·

χ

u. In this case,

(

s

)

isT-invariant corresponding to the support function

−

u

−

div

(

f

)

and thus

µ

−

+

µ

+

=

−

h0

(

−

1

)

−

h0

(

1

)

usingProposition 12.

Letuminandumaxbe respectively the smallest and the largest weights in whichshas non-trivial support and let

ν

=

umax

−

umin. Note that we can bound

ν

by

ν

≤

ν(λ)

:=

max(h,λ)

−

min(h,λ)

.

LetSbe some set of polyhedral divisors corresponding to some open covering ofX and consider some polyhedral divisorD

∈

S. Now, the