AG Codes from Polyhedral Divisors

A G Co des from P olyhedra l Divisors Nathan Ow en Ilten and Hendrik S¨ uß Octob er 29, 2018 Abstract A description of co mplete normal v arieties with lo w er d imensional torus action has b een giv en in [AHS08], generalizing the theory of toric v arieti es. Considering the case where the act in g torus T has co dimension one, we describe T -in v arian t W eil and Cartier divisors and pr o vide formulae for ca lculating global sections, in tersection n u m b ers, and Euler c haracteristics. As a n application, w e use divisors o n these so- called T -v arieties to define new ev aluati on co des called T -codes. W e find estimate s on their minim um distance using intersectio n theory . This generalizes the theory of toric co des and combines it with A G co des on curv es. As the s implest application of our general tec hniques we look at co des on ruled sur faces co m ing from decomp osable v ector bundles. Already t his construction giv es co des that are b etter than the related pro du ct co de. F ur ther examples sho w that w e can imp r o ve these co des b y constructing more sophisticated T -v arieties. T h ese results su ggest to lo ok fu rther for go o d co d es on T -v ariet ies. 1 In tro duc t ion An imp ortant class of linear co des is the class of Algebraic Geometry Co des, in tro duced b y Goppa in 1981. These codes arise b y ev aluating glo ba l sections o f a line bundle on a curv e ov er F q at a num b er of F q -rational p oints; go o d estimates on the dimension and minim um distance of suc h co des can b e obtained b y using the theorem of Riemann-Ro c h. Suc h co des hav e since b een generalized to higher-dimensional v arieties. It is ho w ev er often difficult to obtain non-trivial estimates on the parameters of su c h co des. One class of v a rieties where non-trivial es timates ha ve b een made is that of toric v arieties, whic h one can des crib e com binatorially . T oric v arieties ha v e b een generalized in [AH06 ] and [AHS08] to so- called T -v arieties, whic h are normal v arieties admitting an effectiv e m -dimensional torus action. T -v arieties can then b e describ ed b y a v ariet y Y of dimension dim X − m along with com binato rial data called a divisorial fan. If the acting torus has co dimension one, Y is t hen a curve. The a im of this pap er is to analyze certain ev aluation co des on suc h v arieties; w e shall call these co des T -co des. In short, a T -co de ov er F q is constructed f rom: • a curve Y o ve r F q ; 1 • a so-called divisorial p olytop e (cf. definition 3.8), essen tially a concav e function h ∗ :  h → Div Q Y where  h is a p olytop e with v ertices in some lattice M ∼ = Z m and h ∗ satisfies some additional conditions; • and a set P = { P 1 , . . . , P l } of F q -rational p oin ts on Y . Assuming t ha t the supp ort of h ∗ ( u ) is disjoin t from P for eac h u ∈  h ∩ M , w e can define the T -co de C ( Y , h ∗ , P ) as the sum of a n umber o f pro duct co des: C ( Y , h ∗ , P ) := X u ∈  h ∩ M C u ⊗ C ( Y , h ∗ ( u ) , P ) where C u is the [( q − 1) m , 1 , ( q − 1) m ] co de generated b y ( t u ) t ∈ ( F ∗ q ) m and C ( Y , h ∗ ( u ) , P ) is the A G co de corresp onding to the curv e Y , divisor h ∗ ( u ), and p o int set P . By in terpreting C ( Y , h ∗ , P ) as the image unde r a linear map of the Riemman-Ro ch space of a divisor on a T - v ariet y , we are able to giv e non-trivial estimates for the dimension k and minimum distance d o f this co de. W e b egin in section 2 b y recalling the ba sic theory of T - v arieties. W e then pro ceed to describe divisors and in tersection theory on T -v arieties in section 3. In particular, w e describe all T - in v ariant Cartier and W eil divisors combinatorially , calculate the global sections of a T -in v arian t Cartier divisor, and dete rmine exactly when a T -Cartier divisor is (semi-)ample. F urthermore, w e pro vide formulae for calculating in tersection nu m b ers and for the Euler c haracteristic of a line bundle. The t heory of this section is analogue t o that of divisors on toric v arieties and is essen tial for estimating the pa r a meters of t he ev aluation co des w e construct. In section 4, w e define T -co des and show ho w to estimate dimension and minimum distance, pro viding upp er and lo wer b ounds for b oth parameters. W e g iv e sp ecial atten tion to the case of tw o-dimensional T -v arieties, where we provide a b etter low er b ound for the minim um distance. Finally , w e provide a n um b er of examples in section 5. W e first consider T -co des coming from those ruled surfaces corresp onding to a rank t wo decomp osable v ector bundle. In particular, we show that some of these co des hav e b etter parameters tha n those estimated for the pro duct of a Reed-Solo mon and a one-p oin t Goppa co de. In a second example, w e sho w how one can use the Hasse-W eil b ound to impro ve the low er b ound on the minim um distance. This example also sho ws that there are b etter T -co des than those coming from ruled surfaces. In a final example, w e describ e a T - co de ov er F 7 whose parameters ar e as go o d as an y kno wn linear co de. 2 The The o ry of T -V arieties First w e recall some facts and notatio ns from con v ex geometry . Here, N alw ays is a lattice and M := Hom( N , Z ) its dual. The asso ciated Q -vector spaces N ⊗ Q and M ⊗ Q a r e denoted b y N Q and M Q resp ectiv ely . Let σ ⊂ N Q b e a p oin ted con v ex p o lyhedral cone. A p olyhedron ∆ whic h can b e written as a Mink o wski sum ∆ = π + σ of σ and a compact p olyhedron π is said to ha ve σ as its tail cone. 2 With resp ect to Mink o wski a ddition the p olyhedra with tail cone σ form a semigroup whic h we denote by P ol + σ ( N ). Note that σ ∈ Pol + σ ( N ) is t he neutral elemen t of this semigroup and that ∅ is by definition also an elemen t of Pol + σ ( N ). A p olyhedral divisor with tail cone σ on a normal v ariet y Y is a formal finite sum D = X D ∆ D ⊗ D , where D runs o ver all prime divisors on Y and ∆ D ∈ P ol + σ . Here, finite means that only finitely man y co efficien ts differ fro m the tail cone. W e ma y ev aluate a p olyhedral divisor for ev ery elemen t u ∈ σ ∨ ∩ M via D ( u ) := X D min v ∈ ∆ D h u, v i D in order to obtain an ordinary div isor on Lo c D . Here, Lo c D := Y \  S ∆ D = ∅ D  denotes the lo cus of D . Definition 2.1. A p olyhedral divisor D is called Cartier if ev ery ev aluation D ( u ), u ∈ σ ∨ ∩ M , is Cartier. T o a Cartier p olyhedral divisor we asso ciate a M -gra ded k -algebra sheaf and consequen tly an affine sc heme o ver Lo c D admitting a T M -action: ˜ X := ˜ X( D ) := Spec Loc D M u ∈ σ ∨ ∩ M O ( D ( u )) . F rom [AH06] we know that this construction giv es a normal v ariet y of dimension dim N + dim Y admitting a to rus action of T N with L o c D as its go o d quotient. Moreo ve r, fo r every affine normal v ariety X there exists a p olyhedral divisor D such that X = Sp ec Γ( ˜ X( D ) , O ˜ X( D ) ). X and ˜ X coincide if X admits a torus action with a go o d quotien t. Definition 2.2. Let D = P D ∆ D ⊗ D , D ′ = P D ∆ ′ D ⊗ D b e tw o p o lyhedral divisors on Y . 1. W e write D ′ ⊂ D if ∆ ′ D ⊂ ∆ D holds for ev ery prime divisor D . 2. W e define the in tersection of p olyhedral divisors D ∩ D ′ := X D (∆ ′ D ∩ ∆ D ) ⊗ D . 3. W e define the degree of a p olyhedral divisor deg D := X D ∆ D . 4. F or a (not necess arily closed) p oin t y ∈ Y w e define the fibre p olyhedron ∆ y := D y := P y ∈ D ∆ D . 3 5. W e call D ′ a fac e of D and write D ′ ≺ D if D ′ y is a face of D y for ev ery y ∈ Y . Assume D ′ ⊂ D . This implies M u ∈ σ ∨ ∩ M O ( D ′ ( u )) ← ֓ M u ∈ σ ∨ ∩ M O ( D ( u ))) and w e get a dominan t mo r phism ˜ X( D ′ ) → ˜ X( D ). Prop osition 2.3 ( [AHS08 ], Prop. 3.4, Rem. 3 .5 ) . This morph i s m defines a n op en emb e dding if and only if D ′ ≺ D holds . Definition 2.4. Consider a smo oth pro jectiv e curve Y . A fansy d i v i sor is a formal finite sum Ξ = X P ∈ Y Ξ P ⊗ Z suc h that: 1. Ξ P are p olyhedral sub divisions cov ering N Q and sharing a common ta il fa n; 2. Finite means here that for all but finitely man y p oin ts, Ξ P equals the tail fan. Consider a finite set of p o lyhedral divisors S , suc h that D ≻ D ′ ∩ D ≺ D ′ for ev ery pair D , D ′ ∈ S . Assume furthermore that their p olyhedral co efficien ts D P form the sub divisions Ξ P of a fansy divisor. F rom suc h a set w e ma y construct a sche me ˜ X(Ξ) by gluing X( D )s via ˜ X( D ) ← ˜ X( D ∩ D ′ ) → ˜ X( D ′ ) . Note that w e had to c hec k the co cycle condition, this is done in [AHS08, Thm. 5.3]. F rom theorem 7 .5 ibid. w e kno w t ha t we get a complete v ariet y this w ay . This v ariet y is uniquely determined b y the underlying fansy divisor. Differen t sets S corresp ond to differen t op en co ve rings. Therefore, w e ma y denote t he resulting v ariet y b y ˜ X(Ξ). Theorem 5.6 in [AHS08 ] tell us that for ev ery normal T -v ariet y X with dim X = dim T + 1 w e may find a fansy divisor Ξ and a prop er birat io nal map ˜ X(Ξ) → X . If X admits a go o d quotien t under the torus action this morphism t urns o ut to b e the iden tity . Remark 2.5. F or a fansy divisor Ξ a nd an op en cov ering { U i } i ∈ I of Y w e can find a set S as ab ov e, suc h that for ev ery D ∈ S there is a i ∈ I suc h that Lo c D = U i . Example 2.6. Let Y b e a smo oth pro jectiv e curv e and Q 1 , Q 2 ∈ Y tw o p o in ts. W e consider the fansy divisor Ξ given b y the co efficien ts in figure 1. ˜ X(Ξ) is a complete surface with one dimensional torus action. Example 2.7. W e consider the fansy divisor on P 1 giv en by the co efficien ts in figure 2. ˜ X(Ξ) is a complete (singular) t hr eefold with t w o -dimensional torus action. 4 -3 -2 -1 0 1 2 3 -1 0 1 (a) Ξ Q 1 -3 -2 -1 0 1 2 3 -1 0 1 (b) Ξ Q 2 Figure 1: The fansy divisor of a surface -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 (a) Ξ 0 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 (b) Ξ ∞ -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 (c) Ξ 1 Figure 2: The fa nsy divisor of a threefold 3 Divisors and In te r s ection Theory on T -V arieties F rom no w on w e shall only consider torus actions of co dimension one; w e will study them via fansy divisors. 3.1 Cartier divisors Let Σ ⊂ N Q b e a complete p o lyhedral sub division of N consisting of ta iled p olyhedra. W e consider contin uous functions h : | Σ | → Q whic h are affine on eve ry p olyhedron in Σ. Let ∆ ∈ Σ b e a p olyhedron with tail cone δ . Then h induces a linear function h ∆ 0 on δ = tail ∆ b y defining h ∆ 0 ( v ) := h ( P + v ) − h ( P ) f o r some P ∈ ∆. W e call h ∆ 0 the linear part o f h | ∆ . Definition 3.1. An (in tegral) supp ort function on a po lyhedral sub division Σ is a piecewis e affine func tion a s ab ov e with in teger slope and integer translation. T o b e pr ecise: for v ∈ | Σ | and k ∈ N suc h that k v is a lattice point w e ha v e k h ( v ) ∈ Z . The group of supp ort functions on Σ is denoted by SF Σ . Let Ξ b e a divisorial fa n on a curv e Y . F or ev ery P ∈ Y w e get a p olyhedral sub division Ξ P consisting of p olyhedral co efficien ts. W e consider SF(Ξ), the group of forma l sums P P ∈ Y h P P with 1. h P ∈ SF Ξ P a supp o r t function o f the P -slice of Ξ. 2. all h P ha ve the same linear part h 0 . 3. h P differs from h 0 for only finitely man y p oin ts P ∈ Y . We r efer to this fact by c al ling this sum finite and w e omit those summands which e qual h 0 . Definition 3.2. A supp ort function h ∈ SF ( Ξ ) is called principal if h ( v ) = h u, v i + D , with u ∈ M and D is a principal divisor on Y . By h ( v ) = h u, v i + D w e mean that h P ( v ) = h u, v i + a P , where D = P P a P P . 5 If h = P h P P ∈ SF(Ξ) w e c o nsider a co vering { Y i } of Y suc h t ha t P is a principal divisor on the Y i for ev ery P ∈ Y with h P 6 = h 0 , and suc h that ev ery Y i con tains at most one of these p oints . W e ma y find a set S as ab o ve whic h is compatible with this cov ering and induces Ξ. No w w e c ho ose a D ∈ S with Lo c D = Y i and h P 6 = h 0 . h P is an a ffine function o n ev ery p olyhedron in Ξ P so we get − h P | D P ( v ) = h v , u i + a for some u ∈ M and a ∈ Z . A ssume that div( f ) = aP on Y i ; then f · χ u ∈ K ( ˜ X( D )) T defines a T -inv ariant principal divisor H D on ˜ X( D ). These principal divisors fit together to a Cartier divisor D h on ˜ X(Ξ). Here K ( ˜ X( D )) T := L u ∈ M K ( Y ) · χ u ⊃ Γ( ˜ X( D )) denotes the ring of in v arian t rationa l functions on ˜ X( D ). In this wa y the group of in tegra l supp ort functions on Ξ corr esp o nds to that of in v arian t Cartier divisors on ˜ X(Ξ). 3.2 W eil divisors In general there a r e tw o t yp es of T - in v ariant prime divisors, namely those whic h consist 1. of o rbit closures of dimension dim T ; 2. and of orbit closures of dimension dim T − 1. Prop osition 3.3. If D is a p olyhe dr al d i v isor on a curve with tailc one σ , t h er e a r e one-to-one c orr esp ondenc es 1. b etwe en prime divisors of typ e 1 and p airs ( P , v ) with P a p oint on Y and v a v ertex of ∆ P ; 2. b etwe en prime divisors of typ e 2 and r ays ρ of σ with deg D ∩ ρ = ∅ . Pr o of. Consider the quotien t map π : ˜ X → Loc D . In [AH06] the orbit structure of t he fibres of π is describ ed. Thus , w e kno w that faces F ≺ D y corresp ond to T -inv ar ian t sub v arieties of co dimension dim( F ) in π y := π − 1 ( y ). The corresp ondences f ollo w by using this for closed p oints and the generic p oin t , respectiv ely . Remark 3.4. W e ma y also describ e the ideals of prime divisors in terms o f p olyhedral divisors: 1. F or prime divisors of type 1 corresp onding to a v ertex ( P , v ), the ideal is giv en by I P ,v = M u ∈ σ ∨ Γ( Y , O ( D ( u ))) ∩ { f | ord P ( f ) > h v , u i} . 2. F or prime divisors of t yp e 2, the corresp onding ideal is generated b y all m ultidegrees whic h are not orthogo nal to ρ : I ρ = M u ∈ σ ∨ \ ρ ⊥ Γ( Y , O ( D ( u ))) . 6 Prop osition 3.5. L et h = P P h P c orr esp ond to the Cartier div i sor D h on ˜ X ( D ) . The c orr esp onding Weil divisor is given by − X ρ h 0 ( n ρ ) ρ − X ( P, v ) µ ( v ) h P ( v )( P , v ) , wher e µ ( v ) is the smal lest inte ger k ≥ 1 s uch that k · v is a lattic e p oint. This lattic e p oint is a multiple of the primitive lattic e ve ctor n v : µ ( v ) v = ε ( v ) n v . Pr o of. This is a lo cal statemen t, so w e will pass to a sufficien tly small inv aria n t op en a ffine set whic h meets a particular prime divisor. If w e translate this to our com binato rial lang uage and w e consider a prime divisor corresp o nding to ( P , v ) or ρ then w e hav e to c ho ose a p olyhedral divisor D ′ ≺ D ∈ S suc h that v is also a vertex o f D ′ P or ρ is a ra y in tail D ′ , resp ectiv ely . So w e restrict to following tw o (a ffine) cases: 1. D is a p o lyhedral divisor with tail cone σ = 0 and a single p oint ∆ P = { v } ⊂ N as the only non trivial co efficien t. Moreo ver, Y is affine and factorial. In particular, P is a prime divisor with (lo cal) parameter t P . 2. D is the trivial polyhedral divisor with one dimens io nal tail cone ρ ov er an affine lo cus Y . In the first case w e ma y c ho ose Z -Basis e 1 , . . . , e m of N with e 1 = n v . Consider the dua l basis e ∗ 1 , . . . , e ∗ m . By definition ε ( v ) and µ ( v ) are coprime so w e will find a, b ∈ Z suc h that aµ ( v ) + bε ( v ) = 1. In this situation y := t a P χ be ∗ 1 is ir r educible in Γ( O X ) = Γ( O Y )[ y , t ± ε ( v ) P χ ∓ µ ( v ) e ∗ 1 , χ ± e ∗ 2 , . . . , χ ± e ∗ m ] and defines t he prime divisor ( P , v ). W e consider an elemen t t α P χ u with u = P i λ i e ∗ i . The y -order of t α P χ u is ε ( v ) λ 1 + µ ( v ) α = µ ( v )( h u, v i + α ) , b ecause t α P χ u = y ε ( v ) λ 1 + µ ( v ) α ( t − ε ( v ) P χ µ ( v ) e ∗ 1 ) λ 1 a + bα , and ( t − ε ( v ) P χ µ ( v ) e ∗ 1 ) is a unit. In the second case w e choose a Z -basis e 1 , . . . , e m of N with e 1 = n ρ . W e once aga in consider the dual basis e ∗ 1 , . . . , e ∗ m . In this situation Γ( O X ) = Γ( O Y )[ χ e ∗ 1 , χ ± e ∗ 2 , . . . , χ ± e ∗ m ] . No w ( χ e ∗ 1 ) defines the prime divisor ρ on X . F o r a principal divisor f · χ u , the χ e ∗ 1 -order equals the e ∗ 1 -comp onen t of u , i.e. h u, n ρ i . Example 3.6. F or our threefold example we consider D h where h 0 , h ∞ , h 1 are given b y the tropical p olynomials h 0 = 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) h 1 = 1 ⊙ x ( − 1 , 0) ⊕ 1 ⊙ x ( − 1 , 1) ⊕ 0 ⊙ x (0 , 1) ⊕ 0 ⊙ x (1 , 0) ⊕ 0 ⊙ x (1 , − 1) ⊕ 0 ⊙ x (0 , − 1) 7 where w e are using the tropical semi-ring with o p erations ⊕ = min , ⊙ = +. These suppor t functions are pictured in figure 3 . The W eil divisor corresp o nding to D h is P ρ D ρ + 2 D ( ∞ , 0) + 2 D ( ∞ , ( − 1 , − 1)) . This is the an ti- canonical divisor of X := ˜ X(Ξ) [PS08 ]. (a) h 0 (b) h ∞ (c) h 1 Figure 3: Supp or t functions f o r a T - t hr eefold 3.3 Global sections F or a supp ort function h on X w e ma y consider the M - graded vec tor s pa ce of global sections of D h L ( D h ) = M u ∈ M L ( D h ) u := Γ( X , O ( D h )) . The weight set o f L ( D h ) is defined as the set { u ∈ M | L ( D ) u 6 = 0 } . F or a Cartier divisor giv en b y h ∈ T-CaDiv (Ξ) w e will b ound its w eight set by a p olyhedron as w ell as describ e the graded mo dule structure of L ( D ). Consider a supp or t function h = P P h P P with linear part h 0 . W e define its asso ciated p olytop e  h :=  h 0 := { u ∈ M Q | h u, v i ≥ h 0 ( v ) ∀ v ∈ N } and asso ciate a dual function h ∗ :  h → Div Q Y via h ∗ ( u ) := X P h ∗ P ( u ) P := X P min v ert ( u − h P ) P , where min v ert ( u − h P ) denotes the minimal v alue of u − h P on the ve r t ices of Ξ P . Remark 3.7. Let h b e a conca v e supp o r t function. Ev ery affine piece of h P corresp onds to a pair ( u, − a u ) ⊂ M × Z . h ∗ P is defined to b e the coarsest concav e piecewise a ffine f unction with h ∗ P ( u ) = a u . W e can reform ulate this in terms o f the tro pical semi-ring with op eratio n ⊕ = min , ⊙ = +. W e might think of the h P as g iv en b y tropical p olynomials L w ∈ I ( − a w ) ⊙ x w , then  h = con v ( I ) and h ∗ P ( w ) = a w , i.e. Γ h ∗ P is the reflected low er newton b oundary of the tropical p olynomial for h P . Definition 3.8. A divisorial p olytop e h ∗ is a pair consisting of an ordinar y p olytop e  h ⊂ M Q and a concav e piecewise affine function h ∗ :  h → Div Q Y suc h that 8 1. deg h ∗ ( u ) ≥ 0 for all v ertices u of  h , 2. some multiple of h ∗ ( u ) is principal in case of deg h ∗ ( u ) = 0 for a v ertex u . 3.  h is a lattice p olytop e as is con v (Γ h ∗ P ) fo r eac h P ∈ Y . Let  g ,  h ∈ M Q b e p olytop es. F or a ny conca ve piecewise affine functions g ∗ :  g → Div Q Y and h ∗ :  h → Div Q Y w e define their s um g ∗ + h ∗ to be the piecew ise affine conca v e function on  g +  h giv en by ( g ∗ P + h ∗ P )( u ) = max { h ∗ P ( w ) + g ∗ P ( w ′ ) | u = w + w ′ } . Remark 3.9. F or g , h ∈ S F (Ξ), one easily c heck s that  g +  h ⊂  g + h and that g ∗ P ( u ) + h ∗ P ( u ) ≤ ( g + h ) ∗ P ( u ) for all P ∈ Y and all u ∈  g +  h . F urthermore, if h P and g P are con vex , they corresp ond to tropical polynomials f , f ′ . It follo ws then t hat ( g + h ) P corresp onds to f ⊙ f ′ . Its reflected lo we r newton b oundary is exactly the graph of ( g + h ) ∗ P , thus the equalit y ( g + h ) ∗ P = g ∗ P + h ∗ P holds. T o a divisorial p olytop e h ∗ w e migh t a sso ciate a fansy divisor Ξ and supp ort f unction h on Ξ suc h that h ∗ corresp onds to h in the wa y giv en ab ov e. Indeed, to ev ery h ∗ P w e can asso ciate a tropical p o lynomial f := L ( u,a u ) ( − a u ) ⊙ x u , where ( u , a u ) runs o v er the vertice s of Γ ( h ∗ P ) . This polynomial induces via ev aluation a piecew ise affine function and a p olyhedral sub division Ξ P of N . Remark 3.10. If w e remo ve condition 3 from the definition of a divisorial po lytop e (defini- tion 3.8), the asso ciation in the a b ov e paragraph giv es us a Q -Cartier divisor. F or ev ery fansy divisor there exists a smo oth refinemen t, i.e. a fansy divisor Ξ ′ suc h that ev ery Ξ ′ P is a refinemen t of Ξ P and ˜ X(Ξ ′ ) is smo oth [S ¨ uß08]. Eve r y supp o r t function h on Ξ is ob viously a lso a supp ort function on Ξ ′ . Th us, f or a g iven divisorial p olytop e h ∗ w e migh t a lw ay consider a smo oth fansy divisor Ξ and a supp ort function h on it suc h that the asso ciated dual function equals h ∗ . Example 3.11. W e no w revisit our threefold example. Figure 4 show s a sk etch of h ∗ . W e sho w a refinemen t of the fansy divisor in figure 5 whic h giv es a smo oth threefold. Prop osition 3.12. L et h ∈ S F (Ξ) b e a Cartier divisor with line ar p art h 0 . Then 1. The we i g ht s e t of L ( D h ) is a subset of  h . 2. for u ∈  h we have L ( D h ) u = Γ( Y , O ( h ∗ ( u ))) . 9 (a) h ∗ 0 (b) h ∗ ∞ (c) h ∗ 1 Figure 4: h ∗ for a T -threefold -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 (a) Ξ ′ 0 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 (b) Ξ ′ ∞ -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 (c) Ξ ′ 1 Figure 5: A refined p olyhedral divisor Pr o of. By definition of O ( D h ) we hav e Γ( X , O ( D h )) T = { χ u f | div ( χ u f ) − X ρ h 0 ( n ρ ) ρ − X ( P, v ) µ ( v ) h P ( v )( P , v ) ≥ 0 } . But div ( χ u f ) = P ρ h u, n ρ i ρ + P ( P, v ) µ ( v )( h u, v i + ord P ( f ))( P , v ), so for χ u f ∈ L ( h ) w e get the follo wing b ounds: 1. h u, n ρ i ≥ h 0 ( n ρ ) ∀ ρ 2. ord P ( f ) + h u, v i ≥ h P ( v ) ∀ ( P, v ) The first implies that u ∈  h ∩ M , the second that ord P ( f ) + ( u − h P )( v ) ≥ 0 ∀ ( P , v ). Definition 3.13. F or a cone σ ∈ Ξ ( n ) 0 of maximal dimension in the tail fan and a P ∈ Y we get exactly one p olyhedron ∆ σ P ∈ Ξ P ha ving tail σ . F or a g iv en concav e supp ort function h = P h P P W e hav e h P | ∆ σ P = h· , u h ( σ ) i + a h P ( σ ) . The constan t part give s rise to a divisor o n Y : h | σ (0) := X P a h P ( σ ) P . Prop osition 3.14. A T -Cartier divisor h = P h P ∈ T-CaD iv(Ξ) is ( s emi-)ample if a n d only if al l h P ar e strictly c onc ave (c onc ave) and − h | σ (0) is (semi-)a mple f o r al l tail c ones σ , i.e. deg − h | σ (0) = − P P a h P ( σ ) > 0 (or a multiple of − h | σ (0) is princip al). 10 Pr o of. W e first prov e that semi-ampleness follows fro m the ab ov e criteria. Because h is (strictly) concav e the same is true for h 0 . This implies that t he u h ( σ ) are exactly the v ertices o f  h and h ∗ ( u h ( σ )) = h | σ (0). The semi-ampleness for h ∗ ( u ) , u ∈  h ∩ M f ollo ws from the semi-ampleness at the v ertices. Indeed if D , D ′ are semi-ample divisors on Y this is also tr ue fo r D + λ ( D ′ − D ) with 0 ≤ λ ≤ 1. Ev ery v ertex ( u, a u ) of Γ h ∗ P corresp onds to an affine piece o f h P of the fo r m h u, · i − a u . If w e let f b e suc h that div ( f ) = a u P o n L o c D f or some D ∈ S we then hav e D h | ˜ X( D ) = div( f − 1 χ − u ) (see 3.1 o n page 6) . A p oint ( u, a u ) ∈ M × Z is a v ertex of h ∗ exactly if ( k u, k a u ) is a v ertex of ( k · h ) ∗ . Hence, after passing to a suitable multiple of h w e ma y a ssume, that h ∗ ( u ) is base-p oin t f r ee with f b eing a g lobal section whic h generates O ( h ∗ ( u )) on Lo c D . Th us f χ u is a global section of O ( D h ) whic h generates O ( D h ) | ˜ X( D ) . T o sho w t he other direction, i.e. that semi-ampleness implies the ab o ve criteria, assume that h P is not conca v e. Then this is true also for ev ery m ultiple of ℓ · h P and hence there is an affine piece h u, · i − a u of ℓ h P suc h that a u > ( ℓh P ) ∗ ( u ). This means there is no global section f χ u suc h that div( f ) = a u P . But this con tradicts the base-p oin t freeness of D ℓh and hence the semi-ampleness o f D h . T o get the statement for ampleness note tha t a supp ort function h on a p olyhedral sub division is strictly conca ve if and only if for ev ery supp ort function h ′ there is a k ≫ 0 suc h that h ′ + k h is concav e. Corollary 3.15. ˜ X(Ξ) is pr oje ctive if a n d only if al l Ξ P ar e r e gular sub divisions, i.e. admit a strictly c onvex supp ort function. Remark 3.16. W e see from prop osition 3.14 t ha t fo r h ∈ S F (Ξ), if the T - in v arian t divisor D h is semi-ample, t he corresp onding dual function h ∗ is in fact a divisorial p olytop e. Con- v ersely , if h ∗ is a divisorial p olytop e, the asso ciated divisor o n the asso ciated T -v ariet y is semi-ample. 3.4 In tersection num b ers Definition 3.17. F or a divisorial p olytop e h ∗ w e define its volume to b e v ol h ∗ := X P Z  h h ∗ P v ol M R F or divisorial p o lytop es h ∗ 1 , . . . , h ∗ k w e define their mi x e d volume b y V ( h ∗ 1 , . . . , h ∗ k ) := k X i =1 ( − 1) i − 1 X 1 ≤ j 1 ≤ ...j i ≤ k v ol ( h ∗ j 1 + · · · + h ∗ j i ) Prop osition 3.18. Assume that o n X Ko dair a’s V anishing The or em holds. 1. If D h is semi- ample, for the self-interse ction numb er we get ( D h ) ( m +1) = ( m + 1)! v ol h ∗ . 11 2. L et h 1 , . . . , h m +1 define s e mi-ample div i s ors D i on X (Ξ) . Then ( D 1 · · · D m +1 ) = ( m + 1)! V ( h ∗ 1 , . . . , h ∗ m +1 ) . Pr o of. If w e apply (1) to ev ery sum of divisors f rom D 1 , . . . , D m +1 w e get (2) b y the multi- linearit y and symmetry of in t ersection n umbers. T o pro ve (1) w e first recall that ( D h ) m +1 = lim ν →∞ ( m + 1)! ν m +1 χ ( X , O ( ν D h )) , but for pro jectiv e X := X(Ξ) a nd nef divisors the ranks o f higher cohomology gr o ups are asymptotically irrelev an t [D em01 , Thm. 6.7.] so we get ( D h ) m +1 = lim ν →∞ ( m + 1)! ν m +1 h 0 ( X , O ( ν D h )) . Note that ( ν h ) ∗ ( u ) = ν · h ∗ ( 1 ν u ). Now w e can b ound h 0 b y X u ∈ ν  h ∩ M  deg ⌊ ν h ∗  1 ν u  ⌋ − g ( Y ) + 1  ≤ h 0 ( O ( ν D h )) ≤ X u ∈ ν  h ∩ M deg ⌊ ν h ∗  1 ν u  ⌋ + 1 . (1) On the one hand w e ha v e lim ν →∞ ( m + 1)! ν m +1 X u ∈ ν  h ∩ M deg ⌊ ν h ∗  1 ν u  ⌋ = lim ν →∞ ( m + 1)! ν m X u ∈  h ∩ 1 ν M 1 ν deg ⌊ ν h ∗ ( u ) ⌋ = ( m + 1 ) ! Z  h h ∗ v ol M R . On the other hand, fo r any constant c w e hav e lim ν →∞ 1 ν m +1 X u ∈ ν  h ∩ M c = c · lim ν →∞ #( ν ·  h ∩ M ) ν m +1 = 0 . Th us, if we pass to the limit in (1), t he term in the middle has to con verge to vol h ∗ . Remark 3.19. The theorem allow s us to compute in tersection n um b ers in c haracteristic 0 as we ll as on T -surfaces in p ositiv e c haracteristic b ecause Ko daira’s v anishing the orem holds in thes e cases. W e believ e that the theorem ho lds as w ell for positive c haracteristic in higher dimensions; w ork is b eing do ne to show that the v anishing theorem holds t here. Corollary 3.20. L et h ∈ S F (Ξ) and let C b e any o n e-cycle r ational l y e quivalen t to the interse ction of Cartier divisors, e ach of which c an b e expr esse d as an inte ger line ar c ombi- nation of semi-amp le Cartier divisors. Then D h · C is e qual to D h + P − Q · C for al l p oints P , Q ∈ Y . 12 Pr o of. W e ha ve D h + P − Q · C = ( D h − D − P + D − Q ) · C = D h · C − D − P · C + D − Q · C so it is sufficien t to show that D − P · C = D − Q · C . No w, D − P and D − Q are semi-ample, so w e can apply prop osition 3.18. Using the fact that v ol (( − P ) ∗ + e h ∗ ) = v ol(( − Q ) ∗ + e h ∗ ) for all e h ∈ S F (Ξ) giv es the desired equality . Example 3.21. W e kno w by prop osition 3.14 that D h in o ur threefold is ample. W e ha v e v ol h ∗ = 21. Hence, X is F a no o f degree 21 . 3.5 Gen us of Curv es on S urfaces Let X = ˜ X (Ξ) b e a tw o-dimensional T -v ariet y and let h ∈ S F (Ξ) b e a supp or t function on Ξ. F or a n y curv e C ∈ | D h | , we show ho w to calculate the arithmetic g enus g ( C ). As a corollary , w e can calculate the Euler c har a cteristic χ ( X , O ( D h )) if X is smo oth. Definition 3.22. F or an y h ∈ S F (Ξ) , let in t h ∗ P := X u ∈  ◦ h ∩ M # { a ∈ Z ≥ 0 | a < | h ∗ P ( u ) |} · h ∗ P ( u ) | h ∗ P ( u ) | for eac h p oin t P ∈ Y , where  ◦ h is t he interior of  h . F urthermore, let in t h ∗ := X P ∈ Y in t h ∗ P . Definition 3.23. F or an y h ∈ S F (Ξ) , let # h ∗ P := X u ∈  h ∩ M ⌊ h ∗ P ( u ) ⌋ for an y p oint P ∈ Y and let # h ∗ := X u ∈  h ∩ M deg ⌊ h ∗ ( u ) ⌋ = X Y ∈ P # h ∗ P . Remark 3.24. Note that int h ∗ P is the num b er of “in terio r ” lattice p oin ts b etw een the gr a ph of h ∗ P and 0 counte d with their signs, where lattice p oin ts in heigh t 0 are counted as long as they aren’t on t he b oundary of  h . Similarly , if # h ∗ P ( h ) ≥ 0 for all u ∈  h , # h ∗ P is the sum of the n umber of lattice p oints b et we en the graph of # h ∗ P and 0, where w e coun t no la ttice p oints in heigh t 0 but all lattice p o in ts lying on the graph of h ∗ P . W e will use the f o llo wing lemma: Lemma 3.25. With notation as ab ove, 2 · vol h ∗ P = in t h ∗ P + # h ∗ P for al l P ∈ Y . It fol lows in p articular that 2 · v ol h ∗ = in t h ∗ + # h ∗ . 13 Pr o of. Fix some P ∈ Y . Supp ose now that h ∗ P ( u ) ≥ 0 for all u ∈  h and set ∆ = conv {{ ( u, h ∗ P ( u )) } ∪ { ( u, 0 ) }} , where u ∈  h . This is a conv ex p olytop e in M ′ Q , where M ′ = M × Z . Pick ’s theorem tells us that 2 · v ol ∆ + 2 = #(∆ ∩ M ′ ) + #(∆ ◦ ∩ M ′ ). Now v o l ∆ = vol h ∗ P , #(∆ ∩ M ) = # h ∗ P + #(  h ∩ M ), and #(∆ ◦ ∩ M ) = in t h ∗ P − # (  h ∩ M ) + 2, so the desired equality follo ws. F o r general h ∗ P , c ho ose j suc h that e h ∗ P ( u ) := h ∗ P ( u ) + j ≥ 0 for all u ∈  h . The n 2 · v ol e h ∗ P = in t e h ∗ P + # e h ∗ P and for j ∗ P ( u ) := j we hav e 2 · vol j ∗ P = in t j ∗ P + # j ∗ P . Since v ol , in t , and # are additive at least for integer-v alue d functions, the desired equality f ollo ws for h ∗ P = e h ∗ P − j ∗ P . W e are no w able to prov e t he follo wing prop osition: Prop osition 3.26. L et h ∈ S F (Ξ) b e any supp ort function such that D h is semi-a m ple. Then fo r C ∈ | D h | , the arithmetic genus of C is given by g ( C ) = in t h ∗ + 1 + v ol  h · ( g ( Y ) − 1) , wher e g ( Y ) is the genus of Y . Pr o of. Without loss o f generality , w e can t a k e the curv e C to equal D h . Indeed, arithmetic gen us is inv ariant under ra tional equiv alence and since | D h | isn’t empt y , it m ust con tain some T -inv ariant effectiv e divisor. W e compare the gen us o f C with that of a comparable curv e C 0 on X 0 := Y × P 1 and then compute the genus o f C 0 directly . T o b egin with, note that we can find monoidal tr ansformations π i : X i → X i − 1 1 ≤ i ≤ k suc h that; 1. X i is a T - v ariet y ; 2. π i is T -equiv arian t; 3. There is a biratio na l T - equiv arian t morphism ϕ : X k → X . This is done as fo llo ws: Let Σ b e the fan { Q ≥ 0 , Q ≤ 0 , { 0 }} . . . and let Ξ 0 P := Σ for all p oints P ∈ Y . Then X 0 = ˜ X (Ξ 0 ). Eac h morphism π i corresp onds to an additional sub division in the fan Ξ i − 1 at exactly one p oin t. Th us, w e k eep on refining un til w e get a Ξ k whic h is a smo o t h common refinemen t of Ξ and Ξ 0 ; this gives us our morphism ϕ . Finally , w e let π : X k → X 0 b e the comp osition of the π i ’s. W e now pull back C to C k := ϕ ∗ ( C ). Th us w e no w hav e C k = D h , where h is now considered as a supp or t function o n Ξ k . F urthermore, this do esn’t c hange the arithmetic gen us, that is, g ( C ) = g ( C k ). Define now inductiv ely C i − 1 = π i ∗ ( C i ) for 1 ≤ i ≤ k . One easily c hec ks that C 0 = D e h , where e h ∈ S F (Ξ 0 ) is the supp ort function given b y the divisorial p olytop e e h ∗ P := max u ∈  h h ∗ P ( h ) with  e h :=  h . Note that since C is semi-ample, each C i is semi-ample as w ell. W e will no w calculate the difference b et we en g ( C k ) and g ( C 0 ). W e first consider a sp ecial case, namely , supp ose that h ∗ P is trivial ev erywhere except for at t wo p oints Q 1 6 = Q 2 . If Y = P 1 , all the v arieties X i and X are toric. In this case, t he divisor D h can b e understo o d in toric terms as the p olytop e ∆ h := con v Γ h ∗ Q 1 ∪ Γ − h ∗ Q 2 14 and D e h corresp onds to ∆ e h , whic h is defined in a similar manner. Then g ( C k ) − g ( C 0 ) = I (∆ h ) − I (∆ e h ) , where I ( ∆) is the n um b er of inte rior lat t ice p oin ts of ∆, see for example [LS06], prop. 5.1. But w e hav e I (∆ h ) = int h ∗ Q 1 + in t h ∗ Q 2 − #(  ◦ h ∩ M ) and a similar equation for e h , whic h leads to g ( C k ) − g ( C 0 ) = in t h ∗ − int e h ∗ . (2) No w, equation (2) actually holds in general, not just in the toric case. T o see this, note that fo r eac h 1 ≤ i ≤ k , C i = π ∗ i ( C i − 1 ) + r i · E i , where E i is the exceptional divisor of π i . Then similar to [Har 7 7], V.3 .7 w e ha ve g ( C i ) = g ( C i − 1 ) − 1 2 r i ( r i + 1). Thus, g ( C k ) − g ( C 0 ) = k X i =1 − 1 2 r i ( r i + 1) . Ho we ver, for eac h 1 ≤ i ≤ k , the in teger r i can b e de termined com binatorially by comparing the p olyhedral sub divisions Ξ i P and Ξ i − 1 P for the single p oint P ∈ Y where these f ansy divisors differ. Th us, t he in t egers r i can b e calculated exactly as if we w ere in the toric case, so we get k X i =1 − 1 2 r i ( r i + 1) = in t h ∗ − int e h ∗ . Equation (2) follow s. W e no w calculate g ( C 0 ). F rom the adjunction form ula, w e hav e g ( C 0 ) = D 2 e h + D e h · K 0 2 + 1 for K 0 a canonical divisor on X 0 , see [Ha r 77], V.1.5. The t heorem of Riemann-Ro c h for surfaces ([Har77], V.1.6) giv es us χ ( X 0 , O ( D e h )) = D 2 e h − D e h · K 0 2 + χ ( X 0 , O X 0 ) . Th us, g ( C 0 ) = D 2 e h + 1 + χ ( X 0 , O X 0 ) − χ ( X 0 , O ( D e h )) . No w, χ ( X 0 , O X 0 ) = 1 − g ( Y ) (see [Har77 ], V.2.5) . Lik ewise, if p : X 0 → Y is the pro jection, w e hav e χ  X 0 , O ( D e h )  = χ  Y , p ∗ O ( D e h )  = X u ∈  h ∩ M χ ( Y , O ( e h ∗ ( u ))) = # e h + (1 − g ) · (v ol  h + 1) , 15 where the last equation f ollo ws f rom Riemann-Ro c h for curv es. W e also hav e that D 2 e h = 2 · vol e h . Making these substitutions results in g ( C 0 ) = 2 · v ol e h + 1 + v ol  h · ( g ( Y ) − 1) − # e h = int e h + 1 + v ol  h · ( g ( Y ) − 1) , the second equalit y coming from lemma 3.25. Com bining this with equation (2) completes the pro of. Corollary 3.27. F or any semi-a m ple T -invariant C a rtier divisor D h on a smo oth T -variety X , we ha ve χ ( X , O ( D h )) = # h ∗ − ( g ( Y ) − 1) · #(  h ∩ M ) = X u ∈  h ∩ M χ ( Y , O ( h ∗ ( u ))) . Pr o of. Using the adj unction formula and the Riemann-Ro c h theorem for surfaces as in the ab ov e theorem g iv es us the formula χ ( X , O ( D h )) = D 2 h + 1 + χ ( X , O X ) − g ( C ) for some C ∈ | D h | . W e can use the ab ov e prop osition to calculate g ( C ) . Com bining this with the facts that D 2 h = 2 · v ol h and χ ( X , O X ) = 1 − g ( Y ) along with lemma 3.25 completes the pro of o f the first equalit y . The second equalit y follows dir ectly from the theorem of Riemann-Ro ch for curv es. A t the and of this section w e revisit our surface example and study all introduced concepts at it. Example 3.28. W e lo ok at the Cartier divisor D h on our surface example where h Q 1 and h Q 2 are giv en b y the tropical p olynomials 0 ⊕ ( − 2) ⊙ x 4 and 0 ⊕ ( − 2) ⊙ x 2 ⊕ ( − 1) ⊙ x 3 ⊕ 1 ⊙ x 4 , resp ectiv ely . One easily sees that  h = [0 , 4], and tha t h ∗ Q 1 and h ∗ Q 1 resp ectiv ely corresp o nd to the t ropical p olynomials x 1 / 2 and x ⊕ 4 ⊙ x − 1 ⊕ 7 ⊙ x − 2 . In o ther w ords, h ∗ Q 1 ( u ) = u / 2 and h ∗ Q 2 ( u ) =    u if u ≤ 2 4 − u if 2 ≤ u ≤ 3 7 − 2 u if u ≥ 3 . In figure 6 we sk etch h and the corresp onding divisorial p olytop e h ∗ . W e can use prop osition 3.5 to compute the corr esp o nding W eil divisor: 4 D Q ≤ 0 + 4 D ( Q 2 , 2) + 7 D ( Q 2 , 1) . D h is semi-ample, so b y pro p osition 3 .18 w e get ( D h ) 2 = 15. Finally , from prop o- sition 3.26 w e know that a section of D h has gen us 5 + 4 · g ( Y ). W e ma y a lso start with h ∗ and tak e the dual h to construct a fansy divisor as describ ed ab ov e. W e reco v er Ξ this wa y . X := ˜ X(Ξ) is not sm o oth, but a refinemen t of the polyhedral sub divisions (see figur e 7) giv es a smo ot h surface X ′ (this is will not b e prov ed here; c.f. [S ¨ uß08]). Using corollary 3.27, w e can calculate that χ ( X ′ , O ( D h )) = 12 − 5 · g ( Y ) . 16 -3 -2 -1 0 1 2 3 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 (a) h Q 1 -3 -2 -1 0 1 2 3 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 (b) h Q 2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 (c) h ∗ Q 1 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 (d) h ∗ Q 2 Figure 6: h and h ∗ for a T - surface -3 -2 -1 0 1 2 3 -1 0 1 (a) Ξ ′ Q 1 -3 -2 -1 0 1 2 3 -1 0 1 (b) Ξ ′ Q 2 Figure 7: A refined fansy divisor 17 4 T -C o des and their P arameters 4.1 Construction Let Y b e a curve ov er F q and let h ∗ b e a divisorial p olytop e. Let P = { P 1 , . . . , P l } b e some subset o f the F q -rational p oin ts of Y such that for i = 1 , . . . , l , h ∗ P i is affine and h ∗ P i ( u ) ∈ Z for u ∈  h ∩ M . Let Ξ b e the fansy divisor asso ciated to h ∗ and let Ξ ′ b e some minimal refinemen t suc h that X := ˜ X (Ξ ′ ) is smo oth. Note that for eac h p oin t P i ∈ P , Ξ ′ P i = v ( P i ) + Σ, for a unique la t t ice p oint v ( P i ) and tail fan Σ. Set m = dim M . F or eac h p oint P i let P 1 i , . . . , P ( q − 1) m i b e the ( q − 1 ) m F q -rational p oints o n X of the op en T - orbit con tracting to P i . The supp ort function h asso ciated to h ∗ corresp onds to a semi-ample T -inv ariant F q - rational Cartier divisor D h on X . W e denote the corresp onding line bundle b y O ( D h ) and let L ( D h ) = Γ( X, O ( D h )). F or eac h p o int P j i fix some isomorphism O ( D h ) P j i ∼ = F q . Consider the F q -linear map ev : L ( D h ) → F l ( q − 1) m q f 7→  f P 1 1 , f P 2 1 , . . . , f P ( q − 1) m l  where f P j i is the image of f in F q follo wing the iden tification with O ( D h ) P j i . In other w ords, the ab ov e map ev aluates the rational function f at the l ( q − 1) m p oints P j i 1 ≤ i ≤ l , 1 ≤ j ≤ ( q − 1) m . The image of ev is a linear subspace of F l ( q − 1) m q and thus a linear co de of length n = l ( q − 1) m ; w e denote it b y C ( Y , h ∗ , P ). If P is maximal, w e simply denote it by C ( Y , h ∗ ). Note that although C ( Y , h ∗ , P ) indeed dep ends on the w ay w e identify O ( D h ) P j i with F q , its length n , dimension k , and it s minim um distance d do not. Th us, w e will alw a ys assume that some suc h isomor phisms are giv en, but will not concern o urselv es further with them. Remark 4.1. If h ∗ P i = 0 for i = 1 , . . . , l , then C ( Y , h ∗ , P ) is equiv a lent as co de to the image of t he map ev : M u ∈  h ∩ M Γ ( O ( h ∗ ( u ))) χ u → F l ( q − 1) m q g χ u 7→  g ( P 1 ) χ u ( Q 1 ) , g ( P 1 ) χ u ( Q 2 ) , . . . , g ( P l ) χ u ( Q ( q − 1) m )  where Q 1 , . . . , Q ( q − 1) m are the F q -rational p oints of the m - dimensional torus. Thus , in this case the isomor phisms O ( D h ) P j i ∼ = F q are not only irrelev an t but also unnecessary . Now let C u b e the [( q − 1) m , 1 , ( q − 1) m ] co de generated by ( t u ) t ∈ ( F ∗ q ) m and let C ( Y , h ∗ ( u ) , P ) b e the A G co de corresp onding to the curv e Y , divisor h ∗ ( u ), a nd p oin t set P . Then as mentioned in the in tro duction, w e can also define C ( Y , h ∗ , P ) simply as C ( Y , h ∗ , P ) = X u ∈  h ∩ M C u ⊗ C ( Y , h ∗ ( u ) , P ) . 18 4.2 Estimate on Dimension Assume that the map ev is injectiv e. This is alw ays the case if the b ound g iv en b elow f or the minim um distance is larger tha n zero. W e then ha ve that k = dim F q L ( D h ) . Using pr o p osition 3.12, w e th us get that k = X u ∈  h ∩ M dim Γ( Y , O ( h ∗ ( u ))) . W e can appro ximate k using only the com binatorics of h ∗ . Let γ ( u ) =    deg ⌊ h ∗ ( u ) ⌋ + 1 − g ( Y ) if deg ⌊ h ∗ ( u ) ⌋ + 1 − g ( Y ) > 0 1 if deg ⌊ h ∗ ( u ) ⌋ + 1 − g ( Y ) ≤ 0 and h ∗ ( u ) ≥ 0 0 if otherwis e. Prop osition 4.2. If the e v a luation map ev is inje ctive, then # h ∗ + #(  h ∩ M )(1 − g ) ≤ X u ∈  h ∩ M γ ( u ) ≤ k ≤ # h ∗ + #(  h ∩ M ) . (3) F urthermor e, k = # h ∗ + #(  h ∩ M )(1 − g )) (4) if deg h ∗ ( u ) > 2 g ( Y ) − 2 for al l u ∈  h ∩ M . Pr o of. The leftmost inequality in (3) follows from the definition of γ ( u ). W e no w con- sider the second inequalit y in (3). Fix some degree u ∈  h ∩ M . Then w e alwa ys hav e dim Γ( Y , O ( h ∗ ( u ))) ≥ 0, a nd if h ∗ ( u ) is effectiv e, then dim Γ( Y , O ( h ∗ ( u ))) ≥ 1. Using t he theorem of Riemann-Ro c h (see for example [Har77]) w e also hav e dim Γ( Y , O ( h ∗ ( u ))) ≥ deg h ∗ ( u ) + 1 − g and the inequalit y follo ws. If deg h ∗ ( u ) > 2 g ( Y ) − 2 then equalit y holds, so (4) follo ws. Finally , the righ t inequalit y in (3) follows from dim Γ( Y , O ( h ∗ ( u ))) ≤ deg h ∗ ( u ) + 1. 4.3 General Lo w er Bound on Minim um Distance One strategy to get an estimate for d is using techniq ues of interse ction theory , as first presen ted in [Han01]. These tec hniques ha ve b een applied to toric v arieties, see for example [Han02] and [Rua07]. W e first consider the general case and then sp ecialize to surfaces. Let e ∗ 1 , . . . , e ∗ m b e a basis f o r M . F or P ∈ P and η 1 , . . . , η m − 1 ∈ F ∗ q define l ( q − 1) m − 1 curv es C P ,η 1 ,...,η m − 1 := ( P , v ( P )) ∩ V  { χ e ∗ i − η i } m − 1 i =1  . Eac h p oin t P j i lies on exactly one of t hese curv es. F urthermore, each curv e C P ,η 1 ,...,η m − 1 is rationally equiv alen t to C P := ( P , v ( P )) ∩ V  { χ e ∗ i } m − 1 i =1  = D 0 − P · ( D − e ∗ 1 ) ≥ 0 · . . . · ( D − e ∗ m − 1 ) ≥ 0 19 where the second equalit y follo ws fr om prop osition 3.5, e ∗ i is considered as an elemen t of S F (Ξ), and ( D − e ∗ i ) ≥ 0 is t he effectiv e part of D − e ∗ i . Fix some section s ∈ L ( D h ); this corresp onds to an effectiv e divisor ( s ) 0 = D h + ( s ) . By Z ( s ) w e denote the n umber of p oin ts P j i suc h that s P j i = 0. Equiv alen tly , Z ( s ) is the n um b er of p oin t s P j i con tained in the supp ort of ( s ) 0 . Thus , one has the fo llo wing low er b ound for the minim um distance: d ≥ l ( q − 1) m − max s ∈ L ( D h ) Z ( s ) . Let ( s ) 0 v anish on exactly λ of the curv es { C P ,η 1 ,...,η m − 1 } . F ollowin g [Han01] and setting C = C P for some P ∈ P w e then ha v e that Z ( s ) ≤ λ ( q − 1) + ( l − λ ) D h · C (5) since ( s ) 0 ∼ D h and it follows from corollary 3.20 that D h · C = D h · C P i = D h · C P i ,η 1 ,...,η m − 1 for all 1 ≤ i ≤ l . Assuming that K o daira’s v anishing theorems holds o n X , w e can use prop osition 3.18 to calculate D h · C . W e now b o und λ in a metho d similar to [Rua0 7]. F or the divisorial p olytop e h ∗ :  h → Div Q Y let pr (  h ) be the pro jection of  h to M / Z e ∗ m and define pr( h ∗ ) : pr(  h ) → Div Q ( Y ) b y pr( h ∗ ) P ( u ) = max ( u,u m ) ∈  h ∩ M h ∗ P (( u, u m )) . One easily c hec ks that pr( h ∗ ) is a divisorial p olytop e. Assume that  h ⊂ e u + { u ∈ M | 0 ≤ u i ≤ q − 2 } for some e u = ( e u 1 , . . . , e u m ) ∈ M . This also then ho lds for pr(  h ). W e can write s = χ e u m e ∗ m ·  s 0 + s 1 χ e ∗ m + s q − 2 χ ( q − 2) e ∗ m  where s i ∈ K ( Y )( χ u 1 , . . . , χ u m − 1 ). In fa ct, one easily c hec ks that s i ∈ L ( D pr( h ) ), where D pr( h ) is the T -in v arian t Cartier divisor on the m -dimensional T -v ariet y X pr( h ∗ ) o ve r Y b o t h determined b y pr( h ∗ ). If we restrict s · χ − e u m e ∗ m to some curv e C P ,η 1 ,...,η m − 1 w e get a p olynomial s = s 0 + s 1 χ e ∗ m + s q − 2 χ ( q − 2) e ∗ m ∈ F q [ χ e m ] of degree less than or equal to q − 2. If C P ,η 1 ,...,η m − 1 is a curv e where s v anishes , then s has q − 1 zeros, so s ≡ 0 and s i = 0 for 0 ≤ i ≤ q − 2. Th us the section s i ∈ L ( D pr( h ) ) v anish es on the p oin t of X pr( h ∗ ) corresp onding to the tuple ( P , η 1 , . . . , η m − 1 ). It follows that λ ≤ max t ∈ L ( D pr( h ) ) Z ( t ) . Th us, w e can recursiv ely b o und λ until dim( X ) = 2. 4.4 Lo w er B ou n d on Minim um Distance for dim( X ) = 2 W e can provide a mu c h b etter b ound for Z ( s ) when X is a surface. Consider a global section s of O ( D h ) as b efore suc h that ( s ) 0 v anishes on exactly λ of the curv es { C P i } , say C Q 1 , . . . , C Q λ where the Q i are distinct p oints in P . Thus , s ∈ L ( D e h ), where e h = h + P λ i =1 Q i . Since e h and P λ i =1 ( − Q i ) are conca ve, it fo llo ws that h ∗ = e h ∗ + ( P λ i =1 ( − Q i )) ∗ . In particular, w e hav e that deg e h ∗ ( u ) = deg h ∗ ( u ) − λ. 20 Th us, s can only hav e supp ort in the w eigh ts u ∈  ( h,λ ) , where  ( h,λ ) = { u ∈  h ∩ M | deg ⌊ h ∗ ( u ) ⌋ ≥ λ } . It follo ws immediately that λ ≤ max u ∈  h ∩ M deg ⌊ h ∗ ( u ) ⌋ := λ 0 . Ha ving found a go o d bound for λ , w e no w tr y to im pro ve on the upp er b o und for Z ( s ) in equation (5). By c ho osing a generator w e can iden tify the lat t ice N with Z . Then σ − := Q ≤ 0 and σ + := Q ≥ 0 are the t w o ray s in Σ. Eac h of these rays corresp onds to a T - in v ariant divisor. Let µ − and µ + resp ectiv ely b e the co efficien ts of the prime divisors σ − and σ + in ( s ) 0 . W e w ant to find a low er b ound for the sum µ − + µ + . This is easy if s has supp ort only in a single weigh t u , sa y s = f · χ u : In this case, ( s ) is T - in v ariant correspo nding to the supp o rt function − u − div ( f ) and th us µ − + µ + = − h 0 ( − 1) − h 0 (1) using prop osition 3.5. Let u min and u max b e resp ectiv ely the smallest and the largest weigh ts in whic h s has non-trivial supp ort and let ν = u max − u min . Note that w e can b ound ν by ν ≤ ν ( λ ) := max  ( h,λ ) − min  ( h,λ ) . Let S b e some set of p olyhedral divisors corr esp o nding t o some op en cov ering of X and consider some p olyhedral divisor D ∈ S . No w, the divisor σ − or σ + is contained in ˜ X ( D ) if and only if D has resp ectiv ely σ − or σ + as tail cone. If the tail c one of D is σ + , w e can write s = χ u min f − 1 · ( s 0 + s 1 χ + . . . + s ν χ ν ) with f , s 0 , . . . , s ν ∈ O (Lo c D ) and so ( s ) is the sum o f some effectiv e divisor and the T - in v arian t principal divisor ( f − 1 · χ u min ). Thus, us ing propo sition 3.5, we hav e µ + ≥ − h 0 (1) + u min . On the other hand, if the tail cone of D is σ − , we can write s = χ u max f − 1 · ( s 0 χ − ν + s 1 χ − ν +1 + . . . + s ν ) with f , s 0 , . . . , s ν ∈ O (Lo c D ). Thus , using prop o sition 3.5 again, w e ha v e µ − ≥ − h 0 ( − 1) − u max . Com bining these t wo inequalities gives us µ − + µ + ≥ v o l  h − ν ≥ v ol  h − ν ( λ ) , where w e use the easily c heck ed fact that − h 0 ( − 1) − h 0 (1) = v ol  h . No w, each curv e C P in tersects with σ + in one p oin t ; similarly , C P and σ − in tersect in some other p oint. Neither of these p oints is one of the p oin t s P j i at whic h we are ev aluating our sec t io n s . This means that for eac h of the l − λ curv es where w e calculate the num b er of zeros of ( s ) 0 using in tersection num b ers, w e hav e coun ted at least µ − + µ + to o man y p oin ts. F urthermore, we can use prop osition 3.18 to calculate that D h · C = v ol  h . Th us, w e can impro ve equation ( 5) to Z ( s ) ≤ λ ( q − 1) + ( l − λ ) ν ( λ ) . Summing up the results obtained here leads to the follo wing: 21 Prop osition 4.3. L et C ( Y , h ∗ , P ) b e a toric c o de on a two-dimensiona l T -variety. T hen the minimum distanc e of this c o de is b ounde d fr om b elow by d ≥ min 0 ≤ λ ≤ λ 0 [( l − λ )( q − 1 − ν ( λ ))] . Remark 4.4. In the literature c oncerning toric s urf a ce co des, the estimate for the minim um distance often contains a term in v o lving the self-inters ection n umber of one of the curv es C P . In our case, this term do es not help since C 2 P = 0, whic h can b e easily seen us ing propo sition 3.18. How ev er, the correction we mak e using µ + and µ − has a similar effect. 4.5 Upp er Bound on Minim um Distance A simple upp er b ound on the minim um distance of a toric co de is given in [Rua07]. W e adapt this to the case of T -v arieties. This then giv es us a w ay o f testing if the lo we r b ound on minimum distance attained a b ov e is sharp: Prop osition 4.5. L et f ∈ K ( Y ) b e such that f · χ u ∈ L ( D h ) f or al l u ∈ B ∩ M , wher e B is lattic e isomorphic to a lattic e hyp er-r e ctangle with si d e lengths r 1 , . . . , r m , r i ≤ q − 1 . F urthermor e, supp ose that f vanishes at r 0 of the p oints P i ∈ P . Then d ≤ ( l − r 0 ) ·   ( q − 1) m + m X j =1 ( − 1) j X i 1 <... 0, where the resulting surface is t he Hirzebruc h surface H e , a t o ric v ariet y . Co des obtained by ev aluation on the p oints of the torus w ere considered in [Han02], with parameters considerably b etter than t ho se of pr o duct co des. W e wish t o generalize this to bundles o ve r curv es of higher gen us. Consider the rank t wo lo cally free sheaf E = O Y ⊕ O   X Q i ∈ F q ( Y ) α i Q i   for α i ∈ Z and set X = Pro j ( E ). An y r uled surface coming from a decomp osable v ector bundle is isomorphic to suc h a X . F urthermore, X can easily b e described as a T -v ariet y . Let Σ ⊂ Q b e the fan consisting of the cones Q ≤ 0 , Q ≥ 0 , and { 0 } , and let Ξ b e t he fansy divisor with Ξ Q i = α i + Σ. Then o ne can easily confirm that X = ˜ X (Ξ). W e set α = P α i . Consider no w an y semi-ample T -inv ariant Cartier divisor D h on X . Then h 0 is of the form h 0 ( v ) =  u max · v if v ≤ 0 u min · v if v ≥ 0 for some u min , u max ∈ Z with a := u max − u min ≥ 0. It follow s that  h = [ u min , u max ]. F urthermore, for eac h Q i ∈ F q ( Y ), h Q i is of the form h Q i ( v ) = h 0 ( v − α i ) − b i for some b i ∈ Z . Th us h ∗ Q i ( u ) = α i · u + b i . It follows that deg h ∗ ( u ) = α · u + b . 23 -3 -2 -1 0 1 2 3 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 (a) h Q 0 -1 0 1 2 3 4 -1 0 1 2 3 4 5 6 (b) h ∗ Q 0 Figure 8: h and h ∗ for a simple ruled surface As an example, by setting u min = 0, u max = 3, α 0 = 1, b 0 = 2, and all ot her p ossible parameters to 0, we get the ruled surface with h and h ∗ as pictured in figure 8. W e no w consider the co de C ( Y , h ∗ , P ) for an y set P of F q -rational p oints on Y ; note that h ∗ P is a ffine and in teger- v alued on lattice p oin t s f o r any p oint P ∈ P as required. Set l = # P . F or the sak e of simplicit y w e shall assume that u min = 0, α > 0 a nd α i , b i ≥ 0. This ensures that h is in fact semi-ample, i.e. that h ∗ is a divisorial p o lytop e. One easily confirms that λ 0 = b + a · α and that ν ( λ ) =  a if λ ≤ b ⌊ a − λ − b α ⌋ if λ ≥ b. Using pr o p osition 4.3 w e then hav e that d ≥ min { ( l − b − a · α )( q − 1)) , ( l − b )( q − 1 − a ) } . W e can then use corollary 4.6 to b ound d f r o m ab ov e. Indeed, fo r t ∈ Z , 0 ≤ t ≤ a w e ha ve that h ∗ Q j ( u ) ≥ b j + α j t for all t ≤ u ≤ a . Using t he particular cases t = 0 and t = a results in the b ound d ≤ min { ( l − b − a · α + g ( Y ))( q − 1)) , ( l − b + g ( Y ))( q − 1 − a ) } . Th us, w e hav e upp er and lo w er b ounds for d differing by at most g ( Y ) · ( q − 1). W e no w use prop osition 4.2 to find a lo w er b ound for k . W e alw ays hav e that k ≥ ( a + 1)( b + 1 + α · a/ 2 − g ( Y )) where equality holds if b > g ( Y ) − 2. Supp ose no w that b ≤ g ( Y ); set c = ⌈ ( g ( Y ) − b ) /α ⌉ . No w h ∗ ( u ) is effectiv e for every u ∈  h ∩ M , so w e can improv e the b ound on k to k ≥ ( a + 1 − c )( b + 1 + α 2 ( c + a ) − g ( Y )) + c. (7) Note that equality holds if g ( Y ) ≤ 1. 24 Remark 5.1. In the case Y = P 1 and α i , b i = 0 for all p oin ts Q i with the e xception of some p oint Q 0 , X is t he Hirzebruc h surface H α . If w e set P = F ∗ q , w e recov er the results o f [Han02]. Note that the curv es w e use to co ver the p oin ts of the torus are p erp endicular to those used b y Hans en. In our case, these curv es ha v e se lf -in tersection zero, but the adjustmen t w e mak e with µ − and µ + comp ensates for this. W e no w compare these co des to pro duct codes coming f rom a length q − 1 Reed-Solomon and a Goppa co de. A R eed-Solomon co de has parameters [ q − 1 , k 1 , d 1 ] with d 1 = q − k 1 and k 1 ≤ q − 1 . Assume τ ∈ N with τ > g ( Y ) − 1. Then the G oppa co de on Y gotten b y ev aluating a divisor D of degree τ at l r a tional p oints has parameters [ l, k 2 , d 2 ] with k 2 ≥ τ − g ( Y ) + 1 and d 2 ≥ l − τ , see for example ([PHB98], vol. I c h. 10). The resulting pro duct co de C pr od has parameters [ l ( q − 1 ) , k 1 k 2 , d 1 d 2 ]. F or the pro duct co de w e t hus ha ve the estimates k pr od ≥ k est := k 1 ( τ − g ( Y ) + 1) , d pr od ≥ d est := ( q − k 1 )( l − τ ) . W e can then show the following: Prop osition 5.2. F ix some curve Y and assume that l ≥ q + g ( Y ) − 1 . Using notation as ab ove, we c an find h ∗ and P as ab ove such that the estimate d p ar ameters for C ( Y , h ∗ , P ) ar e b etter than those fo r C pr od . Sp e cific al ly, we show that k est ≤ ( a + 1)( b + 1 + α · a/ 2 − g ( Y )) , (8) d est < min { ( l − b − a · α )( q − 1)) , ( l − b )( q − 1 − a ) } . (9) Pr o of. First, supp ose that τ ≥ ( k 1 − 1). W e then set a = k 1 − 1 and c ho ose some α ∈ N suc h that α ( k 1 − 1) ≤ 2 τ and α ( k 1 − 1) is divisible by t wo. Choo se b i ≥ 0 suc h that b = τ − α ( k 1 − 1) / 2. Cho ose a n y set P consisting of l p oin ts. Equalit y in (8) follows immediately and a quick calculatio n sho ws that ( 9) holds as w ell. Supp ose instead that τ < ( k 1 − 1). Set e k 1 = τ − ( g ( Y ) − 1) and e τ = k 1 + ( g ( Y ) − 1). Consid er then the pro duct co de e C pr od obtained as pro duct of the e k 1 -dimensional Reed- Solomon co de and the Goppa co de corresp onding to the divisor e τ Q 0 . Then one easily confirms that t he estimated minim um distance and dimension for e C pr od are greater than or equal to those of C pr od and that e τ ≥ ( e k 1 − 1). Th us, w e reduce to the first case ab o v e. 5.2 A Co de on an Elliptic Curv e The follow ing example illustrates tec hniques tha t can b e used t o refine our estimate for minim um distance. It also demonstrates that there ar e T -co des with better parameters than the those estimated in the previous example. Before w e b egin, we first no t e the fo llowing lemma: Lemma 5.3. L et D h b e a T -invarian t d ivisor on ˜ X (Ξ) , and let s b e a se ction such that ( s ) 0 is not irr e ducible. Then we c an find functions h 1 , h 2 ∈ S F (Ξ) and s 1 ∈ L ( D h 1 ) , s 1 ∈ L ( D h 1 ) such that: 25 1. D h = D h 1 + D h 2 ; 2. ( s ) = ( s 1 ) + ( s 2 ) ; 3. D h i is not r ational ly e quivalent to 0 for i = 1 , 2 . Pr o of. Since ( s ) 0 is not irreducible, we can write it as the sum of tw o nontrivial effectiv e divisors ( s ) 0 = C 1 + C 2 . Since the Picard gr o up is generated b y T -in v ariant divis ors, we can find h ′ 1 , h ′ 2 ∈ S F (Ξ) suc h that C i = D h ′ i + ( s ′ i ) fo r some s ′ i ∈ L ( D h ′ i ), i = 1 , 2 . W e th us hav e D h + ( s ) = D h ′ 1 + ( s ′ 1 ) + D h ′ 2 + ( s ′ 2 ) . No w set s 1 := s ′ 1 , h 1 := h ′ 1 , and s 2 := s/ s 1 , and le t h 2 b e the s upp ort function corresponding to the T -inv ariant divisor D h ′ 2 + ( s ′ 2 ) − ( s 2 ). These supp ort functions and sections clearly fulfill the desired conditions. W e now return to the divisor on the T -surface considered in example 3.28. F or Y ei- ther P 1 or elliptic, w e hav e already noted that D h is semi-ample; this is the same as sa ying that h ∗ is a divisorial p olytop e. Now if Y = P 1 and Q 1 = 0, Q 2 = ∞ , the T - v ariet y asso ciated to h ∗ is in fact toric and h ∗ corresp onds to the p olytop e in Z 2 giv en b y con v { (0 , 0) , (2 , − 2) , ( 3 , − 1) , (4 , 1) , (4 , 2) } . Let P = Y \ { Q 1 , Q 2 } ; the example of C ( P 1 , h ∗ , P ) is considered in [SS08 ], where it is show n using the Hasse-W eil b ound that d ≥ ( q − 1) 2 − 3( q − 1) − 2 √ 2 + 1 for all q ≥ 19. W e now calculate the para meters d and k fo r C ( Y , h ∗ , P ) in the case that Y is an elliptic curv e. In calculating k , note that deg h ∗ ( u ) > 0 fo r u > 0. Th us, in these degrees w e ha v e that dim L ( D h ) u = deg h ∗ ( u ). On the other hand h ∗ (0) = 0 whic h is effectiv e, so dim L ( D h ) 0 = 1. Adding ev erything up we get that k = 8. Prop osition 4.5 giv es us an easy upp er b ound for d . If w e set f := 1, w e hav e that f · χ u ∈ L ( D h ) for u ∈ 0 , 1 , 2 , 3. Indeed, h ∗ ( u ) is effectiv e in these degrees. Th us, it follo ws that d ≤ l ( q − 1) − 3 l . W e now b ound d from b elow. One easily che cks tha t λ 0 = 3. Lik ewise, o ne can easily calculate that ν (0) = 4, ν (1) = 3 , ν (2) = 1, and ν (3) = 0. Now consider some section s suc h that λ = 1. W e claim that we actually mus t hav e tha t ν ≤ 2. The section s cannot ha ve supp ort in weigh t 0 since deg h ∗ (0) − 1 = − 1. F urthermore, s cannot hav e supp ort in w eigh t 1. Indeed, Γ( Y , O ( Q 2 − P )) = 0 f or a n y p o int P 6 = Q 2 , since Y 6 = P 1 . It fo llo ws that for any section s with λ 6 = 0 or with λ = 0 and ν < 4 w e ha v e Z ( s ) ≤ λ ( q − 1) + l ( 3 − λ ); if w e assume that l ≥ q − 1, it fo llo ws that Z ( s ) ≤ 3 l No w consider some section s suc h that λ = 0 and ν = 4; w e will sho w that under certain assumptions w e also hav e Z ( s ) ≤ 3 l . First, supp ose that ( s ) 0 is ir r educible. Then using the Hasse-W eil b ound for singular curv es as stated in [AP96], w e ha v e that t he n umber #( s ) 0 ( F q ) of F q -rational p oints on ( s ) 0 is b ounded ab o ve b y #( s ) 0 ( F q ) ≤ q + 1 + 2 g √ q where g := g (( s ) 0 ) is the a rithmetic genus of ( s ) 0 . Note that this only dep ends on the divisor D h and not o n s . Now , if we require that q ≥ g + p g 2 + 8 2 ! 2 26 it follow s that Z ( s ) ≤ q + 1 + 2 g √ q ≤ ( q − 1)3 . In our case, it follo ws from prop o sition 3.26 tha t g = 9 so the required b ound on q is q ≥ 89. Supp ose on the other hand that ( s ) 0 is not irreducible. Let h 1 , h 2 ∈ S F (Ξ) b e supp ort functions and s i ∈ L ( D i ) i = 1 , 2 sections as in lemma 5.3, ordered suc h t hat v ol  h 1 ≤ v ol  h 2 . It easily follows that ν ( s ) = ν ( s 1 ) + ν ( s 2 ) and b y remark 3.9 we ha ve h ∗ ≥ h ∗ 1 + h ∗ 2 . No w if s 1 only has supp ort in a single degree, ( s 1 ) 0 is T -inv aria nt. Thus w e hav e Z ( s 1 ) = 0 and Z ( s ) = Z ( s 2 ). Indeed, since λ = 0, ( s 1 ) 0 cannot con tain o ne of the curv es C P co v ering the p oin t s of ev aluation, and all other T -inv ariant prime divisors don’t contain any p oin t s of ev aluation. No w note that h ∗ 2 ≤ h ∗ + ( f ) for some f ∈ K ( Y ) . Th us, g (( s 2 ) 0 ) ≤ g (( s ) 0 ) and if ( s 2 ) 0 is ir r educible, the ab ov e argumen t with the Hasse-W eil b o und giv es the desired b ound. If not, we replace h and s by h 2 and s 2 and rep eat the pro cess until we ha ve an irreducible section and th us the desired b ound, or ha v e sections s ′ 1 and s ′ 2 b oth with support in m ultiple weigh ts. W e ha ve no w reduced to the situatio n wh ere h ′ ∈ S F (Ξ) with s ′ ∈ L ( D h ′ ), h ′ ∗ ≤ h ∗ + ( f ), for this s ′ w e ha v e ν = 4, and h ′ and s ′ admit a decomp osition into h ′ 1 , h ′ 2 and s ′ 1 , s ′ 2 suc h as in lem ma 5.3 suc h that b oth sections hav e s upp ort in m ultiple w eigh ts. W e sho w that this is imp ossible. W e first note that since ν = 4, s ′ i m ust hav e supp ort in the largest and smallest w eigh t s of  h ′ i , whic h w e call u max i and u min i , resp ectiv ely . F urthermore, b y adjusting with T -in v arian t principal divisors we can assume that ( f ) = 0, u min i = 0, and h ′ i ∗ (0) = 0 . W e then hav e ( h ′ i ) ∗ Q 1 ( u max i ) < 2 for i = 1 , 2. Indeed, w e must hav e ( h ′ 1 ) ∗ Q 1 ( u 1 ) + ( h ′ 2 ) ∗ Q 1 ( u 2 ) < 2 for u 1 ∈  h ′ 1 and u 2 ∈  h ′ 2 \ { u max 2 } . The claim follows for i = 1 b y setting u 2 = 0; for i = 2 w e jus t switc h the indices. Now , for at least one i ∈ 1 , 2 w e m ust also ha v e ( h ′ i ) ∗ Q 2 ( u max i ) < 0. Indeed, this f ollo ws from ( h ′ 1 ) ∗ Q 2 ( u max 1 ) + ( h ′ 2 ) ∗ Q 2 ( u max 2 ) ≤ − 1 . F or this i , L ( D h ′ i ) u max i = Γ  Y , O ( h ′ i ∗ ( u max i ))  ⊂ Γ ( Y , O ( Q 1 − Q 2 ))) = 0 . This is how ev er imp ossible since we ha d already concluded that s ′ i has supp ort in weigh t u max i . W e hav e thus sho wn that a section s ∈ L ( D h ) with λ = 0 is either irreducible, in whic h case w e can b ound the n umber of ra t ional p oin ts on it using the Hasse-W eil b ound, or it can b e decompo sed in to T -in v arian t comp onen ts and some remaining section, whic h either is irreducible or which has supp ort in w eigh ts differing by at most 3. Th us, if w e require that q ≥ 89 a nd l ≥ q − 1, we ha ve that for any section s ∈ L ( D h ), Z ( s ) ≤ 3 l . Since our upp er b ound already states that d ≤ l ( q − 1) − 3 l , w e get that in fa ct d = l ( q − 1) − 3 l . This marks an improv emen t o v er the estimates for an y of the T -co des considered in the previous example. Indeed, to get the desired estimated minimum distance w e would ha ve to require b = 0 a nd a ≤ 3. Using equation (7 ), one easily che cks that the dimension of the resulting co de is smaller than 8. 27 5.3 A Computational Example W e are a ble to pro vide a T -co de ov er F 7 with parameters [66 , 1 9 , 30 ], whic h is as go o d a s the b est kno wn co de (c.f. [G ra07]). W e set Y = V ( z y 2 + 6 x 3 + 4 z 3 ) ⊂ P 2 F 7 and consider the divisorial p o lytop e giv en in figure 9. Fixing t w o F q -ration p oin t s Q 1 and Q 2 w e can compu te a generator matrix of C ( Y , h ∗ ) using Macaulay 2 [GS08] and the toric codes pac k age [Ilt08]. W e can then compute the minimal distance using Magma [BCP97]. -1 0 1 2 3 4 5 6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 (a) h ∗ Q 1 -1 0 1 2 3 4 5 6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 (b) h ∗ Q 2 Figure 9: A divisorial p o lytop e defining a [66 , 19 , 3 0 ] 7 co de It is easy to see that the length and dimension of C ( Y , h ∗ ) are alw a ys resp ectiv ely 66 and 19. How ev er, the minim um distance can b e either 29 or 30, dep ending on the c hoice of Q 1 and Q 2 . F or example, setting Q 1 = (1 : 2 : 1), Q 2 = (1 : 5 : 1) results in a minim um distance of 30, whereas Q 1 = (1 : 2 : 1), Q 2 = (0 : 1 : 1) r esults in a minimum distance of 2 9. In fact, the a utomorphism group of Y divides the set o f all pair s of rational p o in ts on Y into tw o equally large subsets; using pa ir s in one subset results in a minimum distance of 30, whereas pairs from the ot her subset result in a minimu m distance of 29. F or t his example, w e a r e also able to use prop osition 4.5 t o easily sho w that d ≤ 30. Indeed, it is not difficult to find a section f ∈ Γ( Y , O (3 Q 1 + 3 Q 2 )) v anishing at 6 distinct p oints of Y ( F q ) \ { Q 1 , Q 2 } . Thus , f ∈ L ( D h ) 3 and w e g et d ≤ 66 − 6 · 6 = 30. References [AH06] Klaus Altmann a nd J ¨ urgen Hausen. P olyhedral divisors and a lgebraic to rus actions. Math. Ann. , 33 4 (3):557–6 0 7, 2006. 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[LS06] John L ittle and Hal Sc henc k. T oric surface co des and Minko wski sums. SIAM J. Discr ete Math. , 20(4 ) :999–1014 (electronic), 2006. [PHB98] V. S. Pless, W. C. Huffman, and R. A. Brualdi, editors. Handb o ok of c o ding t he ory. Vol. I, II . North-Holland, Amsterdam, 1998. [PS08] Lars Pe t ersen and Hendrik S ¨ uß. T o r us in v arian t divisors. arXiv:math/0811.05 1 7v1, 2008. [Rua07] D iego R uano. On the parameters of r - dimensional toric co des. Finite Fields Appl. , 13(4):962– 976, 2007. [SS08] Iv an Sopruno v and Jen y a Soprunov a. T oric surface co des and Mink o wski length of p olygons. arXiv:0802.2088v1, 2008. [S ¨ uß08] Hendrik S ¨ uß. Canonical divisors o n T-v arieties. arXiv:math/0811.062 6v1, 2008. Na than Il ten Ma thema tisches Institut Freie Un iversit ¨ at Berlin Arnimallee 3 14195 Berlin, Germany E-mail addr e ss : nilten@cs.uc h icago.edu 29 Hendrik S ¨ uß Institut f ¨ ur Ma thema tik LS Algeb ra und Geometrie Brandenbu rgi s che Tech nische Univer- sit ¨ at Cottb us PF 10 13 44 03013 Cottbus, Germany E-mail addr e ss : suess@math.tu-cottbus.de 30