Vol. 31, No. 2, 2005
On
invariants
of polynomial
functions*
By
Yoshiaki FUKUMA
(Received July 5, 2004)
(Communicated by Kodai Mathematical Journal)
Abstract. In this paper, we define some invariants of polynomial functions which are derivative from invariants of polarized varieties, and we study their properties. As applications, by using their invariants, we study polarized toric varieties, and we also classify finite partially ordered sets.
1. Introduction
Let X be an n-dimensional projective variety over the field of complex numbers and let L be an ample line bundle on X. Then (X, L) is called a polarized variety. If X is smooth, then we say that (X, L) is a polarized manifold.
In order to study polarized varieties, it is important to use their invariants. The Ģ-genus Ģ(L) and the sectional genus g(L) are famous invariants of (X, L). Many people studied these invariants. In particular, T. Fujita studied polarized manifolds by the Ģ-genus, and the sectional genus, and he gave a classification of polarized manifolds with Ģ(L)_??_2 and g(L)_??_2. (See [4] in detail.)
On the other hand, in order to study polarized varieties more deeply, in [5], [6], [7], [9] and [10], for every integer i with 0_??_i_??_n, the author defined the i-th sectional H -arithmetic genus xHi(X, L), the i-th sectional geometric genus gi(X, L), and the i-th Ģ-genus Ģi(X, L) of (X, L).
The reason why we call xHi(X, L) (resp. gi(X, L)) the i-th sectional H-arithme tic genus (resp. the i-th sectional geometric genus) is as follows. Let X be a smooth projective variety of dimension n, let i be an integer with 1_??_i_??_n, and let L be an ample and spanned line bundle on X. Then by the Bertini's theorem, there exists a ladder X=: Xp D Xi D D Xn_i such that Xj is smooth and Xj•¸|Lj-1| for every integer j with 1_??_j_??_n-i, where Lj:= Lj-1|xj. Here we note that dim Xj=n-j. Then we can show that xHi(X , L)=x(Oxn-i) and g(X, L) = h2(Oxn_i). Namely xHi(X, L) (resp. gi(X, L)) is equal to the arithmetic
Key words and Phrases. Polynomial function, polarized varieties, toric varieties, sectional genus, i-th sectional geometric genus, Ģ-genus, i-th Ģ-genus, partially ordered set.
* This research was partially supported by the Grant-in-Aid for Young Scientists (B) (No. 14740018), the Ministry of Education, Culture, Sports, Science and Technology, Japan.
genus of Xn-i in the sense of Hirzebruch (resp. the geometric genus of Xn-i). In view of these facts, we can expect that, by using these invariants, we should be able to show results for polarized varieties which are analogous to results for i - dimensional varieties. In [8] and [10], we consider the case where i=2 and we get some results which are analogous to results in the theory of surfaces. In [5] and [7], we proposed some problems for the case where i=2.
Here we note that g0(X, L) is the degree of L and g1(X, L) is the sectional genus of (X, L). Namely the i-th sectional geometric genus of (X, L) is a generalization of the degree and the sectional genus of (X, L) (see Remark 2.6).
Furthermore if i=1, then Ģ1(X, L)=Ģ(L). So the i-th Ģ-genus is a generalization of the Ģ-genus (see Remark 2.6 (2)).
Moreover we can find that there are some relations between gi(X, L) and Ģi(X, L), and gi(X, L) (resp. Ģi(X, L)) has properties similar to that of g(L) (resp. Ģ(L)) if X is smooth and L is ample and spanned. (See [5], [6], and [9] in detail.)
On the other hand, in [14] A. Ooishi generalized the notion of the Ģ-genus and the sectional genus of polarized varieties to the case of polynomial functions. (For the definition of polynomial functions, see Definition 2.1 below.) He studied properties of these invariants of various polynomial functions. Inspired by [14], in this paper, we first define the notion of the i-th sectional H-arithmetic genus for polynomial functions. Next we define the notion of polynomial functions in two variables associated with polynomial functions (see Definition 3.2), and we define the i-th sectional geometric genus and the i-th Ģ-genus of these (see Definition 3.3). Here we note that if i=1, then the first sectional geometric genus (resp. the first
Ģ-genus) of any polynomial function in two variables associated with a polynomial function h is equal to the sectional genus (resp. the Ģ-genus) of h which was defined by Ooishi. (See Remark 3.4 (1).)
The contents of this paper are as follows. In Section 2, we list up some definitions and results which are used later. In Section 3, first we define the i - th sectional H-arithmetic genus of polynomial functions. Next we define the i-th sectional geometric genus and the i-th Ģ-genus for polynomial functions in two variables associated with polynomial functions, and we study some properties of these. Moreover we show that the i-th sectional H-arithmetic genus (resp. the i-th sectional geometric genus, the i-th Ģ-genus) of (X, L) is the i-th sectional H-arithmetic genus (resp. the i-th sectional geometric genus, the i-th Ģ-genus) of the polynomial function h0(xL) (resp. a polynomial function in two variables associated with h0(xL)). In Theorem 3.3, Corollary 3.3, and Theorem 3.4, we also study invariants of polarized toric varieties. In Section 4, as an application, we classify finite partially ordered sets by these invariants. In [18], Yamamoto studied the sectional genus and the Ģ-genus of the polynomial function h which is defined by the order polynomial of a finite partially ordered set (see Definition 4.1). In particular, she tried to classify finite partially ordered sets for the case where its
sectional genus is small. As the first attempt, she calculated the sectional genus and the Ģ-genus for the following cases:
(1) The case where the order of partially ordered sets is less than or equal
to 4.
(2) Special cases (for example, totally ordered sets).
By this consideration, she proposed the following conjecture.
CONJECTURE
1.1 (Yamamoto). Let P be a finite partially ordered set, and
let h be the polynomial function which is defined by the order polynomial of the
finite partially ordered set. Then gs(h)=0 if and only if P is a totally ordered set.
(Here gs(h) denotes the sectional genus of h (see Definition 2.2).)
Here we note that Yamamoto proved that if P is a totally ordered set, then gs(h)=0. But she was not able to prove that P is a totally ordered set if gs(h)=0.
In Section 4, we study the i-th sectional geometric genus gi(P) and the i-th Ģ-genus Ģi(P) of a polynomial function in two variables associated with the polynomial function which is defined by the order polynomial of P. We note that gs(h)=g1(P) and gĢ(h)=Ģ1(P), where h is the polynomial function which is defined by the order polynomial of P. In Theorem 4.1, we give a criterion for gi(P)=0. By Theorem 4.1 we can prove that the above conjecture is true. Moreover, by using some properties of g2(P) and Ģ2(P), we get a classification of P if g2(P)=0 or g1(P)=1.
The author would like to thank the referee for giving some comments and suggestions.
Notation and conventions.
We say that X is a variety if X is an integral separated scheme of finite type.
In particular X is irreducible and reduced if X is a variety.
Varieties are always assumed to be defined over the field of complex numbers.
The tensor products of line bundles are denoted additively.
For a line bundle L on a projective variety X, hi(L) denotes dim Hi(X, L).
Basically, in this paper, symbols m and n denote elements of the set of integers,
and symbols x, y, and t denote variables unless otherwise mentioned.
C: the set of complex numbers.
R: the set of real numbers.
Q: the set of rational numbers.
Z: the set of integers.
S[t]: the set of polynomials in t whose coefficients belong to the ring S.
Os[t]: the zero element of S[t].
For a real number
m and a non-negative
integer
n, let
Then for n fixed, [t]n and [t]n are polynomials in t whose degree are n. For any
non-negative integer n,
Assume
that
m and n are non-negative
integers.
Then
we put
We note
that
if m_??_0.
2. Preliminaries
DEFINITION 2.1. (1) (See [13], •˜1.) Let f•FZ•¨Z be a function. Then f is called a polynomial function if f satisfies the following.
(A) There exist an integer N1 and a polynomial P(t)•¸C[t] such that f(n)= P(n) for every integer n with n>N1.
(B) There exists an integer N2 such that f(m)=0 for every integer m with
m<N2.
In this case we put Pf(t):=P(t) because P(t) depends on the function f. We call this polynomial Pf(t) the polynomial associated with f.
(2) Let f be a polynomial function. Then
n(f):=min{m•¸Z|f(n)=Pf(n) for •Ín>m}, z(f):=max{leZ( f(n)=O and Pf(n)Ofor.Vn<l}.
(3) Let q(t) E C[t, t-1] and we put c(t) = > a2tz. Then we put d(~b) := max{k I ak 0}.
(4) Let f be a polynomial function such that Pf(t)•‚0C[t]. Then we put d(f):=d(Pf).
REMARK 2.1. (1) Let f(t)•¸C[t]. If f(a)•¸Q for every a•¸Z, then f(t)•¸ Q[t]. (We can easily prove this and we omit the proof.)
(2) Let f(t)•¸C[t] with f(t)•‚0C[t]. If there exists an integer N such that f(n)•¸Z for every integer n with n_??_N, then f(n)•¸Z for every integer n.
PROOF. We prove this by induction on deg f. If deg f=0, then this is true.
Assume that the assertion is true for deg f<k. Next we consider the case where deg f=k. We put h(n):=f(n)-f(n-1). Then deg h=k-1 and there exists an integer N' such that h(n)•¸Z for every integer n_??_N'. Hence by induction hypothesis, we obtain h(n)•¸Z for every integer n.
On the other hand, assume that there exists an integer n with f(n)•¸Z and f(n-1)•¸C•_Z. Then h(n)•¸C•_Z. But this is a contradiction. Hence we get the assertion.
(3) Let f be a polynomial function such that Pf(t)•‚0C[t]. Then Pf(n)•¸Z for every integer n.
PROOF. Since f is a polynomial function, there exists an integer N such that f(n)=Pf(n) for every integer n with n_??_N. Since f(n)•¸Z, we obtain Pf(n)•¸Z for every integer n with n_??_N. Hence by (2) above we get the assertion.
(4) Let f be a polynomial function. By (1) and (3) above, we get Pf(t)•¸Q[t]. NOTATION 2.1. (1) Let f•FZ•¨Z be a function such that f(n)=0 for every integer n with n•á0. We put
(2) PF denotes the set of polynomial functions, and
REMARK 2.2. (1) Let f be a polynomial function. Then ƒ¢f, ƒ¢+f and •Þf are also polynomial functions. (This can be easily proved and we omit the proof.)
(2) Let P(t) be a polynomial in t. Then we set
If f is a polynomial function, then OP f (t) = P1(t) and ©-P1(t) = Po+ f (t).
PROOF. Since Pf(n)=1(n) for n•â0 by definition, we get ƒ¢Pf(n)=ƒ¢f(n)
for any n•â0. Therefore ƒ¢Pf(n)=Pƒ¢f(n) for any n•â0. But since ƒ¢Pf(t) and
PĢf(t) are polynomials, we have ĢPf(t)=PĢf(t).
By the same argument as above, we can prove the second equality. This
completes the proof.
LEMMA 2.1. (1) Let P(t)•¸C[t] be a polynomial in t such that P(t)•‚0C[t] and P(n)•¸Z for any integer n, and let d be the degree of P(t). Then there exists a unique sequence of integers (e0, el,... , ed) such that the following holds.
(2) Let P(t)•¸Z[t] with deg P(t)=d. Then there exists a unique sequence of integers (a0,... , ad) such that
PROOF. These can be easily proved and we omit the proof.
NOTATION 2.2. (1) Let p(t) be a polynomial in t such that p(n)•¸Z for every integer n. We put d=deg p(t). Then by Lemma 2.1, there exists a unique sequence of integers (e0, el,... , ed) such that the following holds.
Here we put ei(p):=ei.
(2) Let f be a polynomial function and let Pf(t) be the polynomial associated with f. (By Remark 2.1 (3), we obtain Pf(n)•¸Z for every integer n.) Then we put ei(f):ei(Pf).
DEFINITION 2.2. (See [14].) Let f•FZ•¨Z be a polynomial function. Then the sectional genus gs(f) of f and the ƒ¢-genus gƒ¢(f) of f are defined as follows.
(1)
(2)
REMARK 2.3. (1) Let f be a polynomial function. Then by Remark 2.2 (1), •Þf is also a polynomial function. By Lemma 2.1 (1) there exists a sequence of integers (eo(Vf),... , ed(o f) (V f)) such that
For every integer m with m•â0, we have P•Þf(m)=•Þf(m). Hence for every integer m with m•â0, we get
On the other hand, since f is a polynomial function, there is a sequence of integers
(eo(f),...,ed(f)(f))
such that
We note that d(•Þf)-1=d(f). Hence we obtain ei(•Þf)=ei(f) for any integer i with 0_??_i_??_d(f)=d(•Þf)-1.
(2) Let f(t) be a polynomial function such that d(f)_??_1. Then we obtain ei(f)=ei(•Þf) for i=0, 1, and d(•Þf)=d(f)+1. Hence in this case, by (1) we
get
There is the following result in [13], Proposition 1.1. But its proof is omitted there. So here we give a proof of this theorem for convenience.
THEOREM 2.1. Let f•FZ•¨Z be a function. Assume that for every positive integer N there exists an integer u such that u_??_N and f(u)•‚0. Then the following are equivalent.
(1) f is a polynomial function.
Moreover, if these conditions are satisfied, then we have (1) 0, d = d(V f) = d(f) +1, and n(f) = d(q5) d.
PROOF. (A) Assume (1). By assumption we obtain Pf(t)•‚0Q[t]. Let m= z(f). We put h(n):=f(n+m). Then h(n)=0 if n<0 and
We also note that Ph(n)=Pf(n+m) and Fh (t) = > > h(n)tn.
Here we put Q(t) = ~n>o h(n)tn ~n>o Ph(n)tn. By Remark 2.1 (3) we get Ph(n)•¸Z for every integer n. We also note that h(n)=Ph (n) for any n•â0. Hence Q(t)•¸Z[t]. Since Ph(t) is a polynomial in t and Ph(n)•¸Z for any integer n, by Lemma 2.1 (1) there exists a unique sequence of integers (e0,... , ed(h)) such that e0•‚0 and
Here we note that
Here we put
and
Then
We note that ƒÓ(t)•¸Z[t, t-1]. Therefore, putting d:=d(h)+1, we get the assertion (2). (We note that d_??_1 and d=d(f)+1 because d(h)_??_0 and d(f)=d(h).)
Assume that Q(t)=0z[t}. Since m=z(f), we get Ph(-1)=Pf(m-1)•‚0 and h(-1)=0. So we get n(h)=-1 because Q(t)=0z[t]. Furthermore 0 ~ Ph(-1) _ (-1)d(h) ed(h). Hence deg R(t)=d(h). Therefore deg R(t)=d(h)=d(h)+1+n(h).
If Q(t)•‚0z[t], then n(h)=deg Q(t) and deg R(t)=d(h)+1+deg Q(t)= d(h)+1+n(h).
In each case, we obtain
Since n(h)=n(f)-m, we obtain d(q) = d(h)+1+n(f) = d(f)+1+n(f) = d+n(f). We also note that q(1) = lmR(1) = eo 0.
(B) Next we assume (2). Then there exists a nonnegative integer l such that t1gp(t) E Z [t]. We put P(t) ;_ tlq(t). Then there exist Q(t), R(t)•¸Z[t] such that P(t) = Q(t)(1 t)d + R(t) with R(t)=0z[t] or deg R(t)_??_d-1.
If R(t)=0Z[t], then Ff(t)tl=Q(t) and f(n)=0 for all n•â0. But this contradicts the assumption. Hence R(t)•‚0Z[t]. We put dr:=deg R(t). We note that 0_??_dr_??_d. By Lemma 2.1 (2), there exists a unique sequence of integers
(ao,...,adr) such that
So we obtain
Here we put
(a) If p>q-1,
then
(b) If p<-1, then f(p)=0.
Therefore f(t) is a polynomial function and we get the assertion (1). REMARK 2.4. Let f be as in Theorem 2.1. If f e PJ''-°, then we get ƒÓf(t)•¸ Z[t]. (See [13], Theorem 1.3 (4).)
NOTATION 2.3. Let f be a polynomial function. Then by Theorem 2.1 there exists ƒÓ(t)•¸Z[t, t-1] such that Ff(t)=ƒÓ(t)/(1-t)d. Then we put cb(t) = ~iEZ aiti. Let ƒÓf(t):=ƒÓ(t) and ai(f):=ai.
PROPOSITION 2.1. Let f be a polynomial function and let d=d(f)+1. If d_??_1, then the following hold.
(1) f(n) = L1i<n ai(f) (n-d--d-1). (2) fln) _ ~i<n ai(f ) (d+d-i)
(Here we use Notation 2.3.)
PROOF. See [13], Theorem 1.3 (2).
REMARK 2.5. We note that there is a typographical error in [13], Theo rem 1.3 (2).
DEFINITION 2.3. Let f(t) be a polynomial function such that f(t)•¸
pF_??_0. Here we use Notation 2.3. Then f(t) is said to be h-semipositive if ai(f)_??_0 for
every integer i.
In Notation 2.4, Definition 2.4, and Remark 2.6 below, let(X, L) be a polarized variety of dimension n.
NOTATION 2.4. Let x(tL) be the Euler-Poincare characteristic of tL. Then x(tL) is a numerical polynomial in t by [11], Chapter I, •˜1, Theorem, p.295. (For the definition of a numerical polynomial, see [11], Chapter I, •˜1, p.295.) Here we
put
DEFINITION
2.4 (See [5], [6], [7], [9], and [10]). For every integer i with
0_??_i_??_n
we define the following:
(1) The i-th sectional H-arithmetic genus xHi(X, L) is defined by the follow
ing:(2) The i-th sectional geometric genus gi(X, L) of (X, L) is defined by the
following:
(3) The i-th Ģ-genus Ģi(X, L) of (X, L) is defined by the following:
REMARK 2.6. (1) If i=0, then g0(X, L)=xH0(X, L)=Ln.
(2) If i=1, then g1(X, L)= g(L) (resp. Ģ1(X, L)=Ģ(L)), that is, g1(X, L) (resp. Ģ1(X, L)) is the sectional genus (resp. the Ģ-genus) of (X, L). (For the
definition of the sectional genus and the Ģ-genus, see [4], Chapter I (2.1) and (2.2).) DEFINITION 2.5. Let M be a subset of Rn.
(1) The intersection of all convex sets which include M is called the convex hull Conv(M) of M.
(2) The intersection of all affine sets which include M is called the afine hull Aff(M) of M.
(3) We define dim M:=dim Aff(M).
(4) If M is a finite set, then P:=Conv(M) is called a convex polytope. Here we note that P is compact.
DEFINITION 2.6. Let P be a convex polytope in Rn.
(1) P is called an integral convex polytope if its vertices all lie in Zn. (2) We put i(P, t):=#(tP•¿Zn), where
tP:={tx|x•¸P}.
THEOREM 2.2 (Ehrhart's theorem). Let P be an integral convex polytope of dimension d in Rn. Then i(P, t) is a polynomial in t and its degree is d.
PROOF. See [2] or [3] Chapter IV, Section 6.
THEOREM 2.3. Let P be an integral convex polytope in Rn. We define a function fp:Z•¨Z as
Then fp(t) is a polynomial function and h-semipositive.
PROOF. By definition and Theorem 2.2, fp(t) is a polynomial function. By
[15] (see also [1], [16]) fp is h-semipositive.
DEFINITION
2.7. Let P be an integral convex polytope in Rn. The polyno
mial function fp(t) in Theorem 2.3 is called the polynomial function associated with
P.
REMARK
2.7. Let P be an integral convex polytope in Rn. Assume that
dim P=n.
Then there exists a toric variety Xp of dim Xp=n and an ample line
bundle Lp on Xp such that x(mLp)=i(P,
m) for every integer m with m_??_0.
3. Invariants of polynomial functions in two variables associated with polynomial functions
DEFINITION 3.1. Let f•FZ•¨Z be a polynomial function, and let Pf(t)•¸ Q[t] be the polynomial associated with f such that Pf(t)•‚0Q[t]. We use Nota tion 2.2. Then, for every integer i with 0_??_i_??_d(f), we define the i-th sectional H-arithmetic genus xHi(f) of f as follows.
EXAMPLE 3.1. Let (X, L) be a polarized variety of dimension n. We put h(t):=h0(tL). Here we note that by the Hirzebruch-Riemann-Rock theorem x(tL) is a polynomial in t. By the Serre vanishing theorem, we get hi(mL)=0 for every pair of integers i and m with 1_??_i_??_n and m•â0. Hence h(m)=x(mL) for m•â0. We also note that h(m)=0 if m<0. Hence h(t) is a polynomial function with Ph(t)=x(tL).
PROPOSITION 3.1. Let (X, L) be a polarized variety of dimension n and let i be an integer with 0_??_i_??_n. Let h(t):=h0(tL). Then xHi(h)=xHi(X, L).
PROOF. First we prove the following claim.
CLAIM 3.1. Let f be a polynomial function with Pf(t)•‚0Q[t]. We use no tations in Definition 2.1 and Notation 2.2. For every integer m with m•â0,
PROOF. Here we put d: = d(Vf) = d(f) + 1 and e3 := e2(D f).
We note that d=d(f)+1_??_1. Then by Lemma 2.1 (1) and Notation 2.2 (2)
On the other
hand
Hence
By Remark 2.3 (1), ej=ej(f)
for 0_??_j_??_d(f). This completes the proof of
We go back to the proof of Proposition 3.1. In this case d(h)=n
and
e3 (PVh) = e~ (h) for every integer j with 0_??_j_??_n. Here we put ej:=ej(h).
We note that Ph(t) = x(tL)
Of[t] because L is ample. By Claim 3.1, we get
for m•â0. We also note that by the Serre vanishing theorem, h(m)=x(mL) for any m•â0. Since x(tL) is a polynomial in t, we get
Since
we get
Hence
for every integer
i with 0_??_i_??_n
Hence we get the assertion of Proposition 3.1.
Proposition 3.1 gives the reason why we call the invariant xHi(h) the i-th sectional H-arithmetic genus of h.
Next we calculate xHi(h) by using aj(h) (see Notation 2.3).
THEOREM 3.1. Let h•FZ•¨Z be a polynomial function with Ph(t)•‚0Q[t]. Then there exists ƒÓh(t)•¸Z[t, t-1] such that Fh(t)=ƒÓh(t)/(1-)d by Theorem 2.1. (Here d=d(h)+1.) We put ch(t) :_ L.+nEZ an(h)tn. Then for every integer i with 0_??_i_??_d(h)
PROOF. By Lemma 2.1 (1) and Notation 2.2 (2), we get
First
we consider
the following:
By repeating the above process, we get deg OPh(t) = d(h) -k for any positive
integer k, and
In particular
(*)
(Here we note that d=d(h)+1.)
Since
On the other hand since Fh(t)(1-t)d=ƒÓh(t), we obtain
Hence Foh(t)(1
t)d-1 = c/h(t). On the other hand, by Remark 2.2 (2)
d(Poh) = d(OPh) = d(h)
1.
By the same argument as above we get Fokh(t)(l-t)d-k = cbh(t)
and d(&h) _
d(Pokh) = d(h)
k = d k 1 for any natural number k. Hence
So we obtain q k (t) = ~bh(t) and a3(LV
h) = a3(h). We note that by Re
mark 2.2 (2), we can prove that
(**)
for any n and any natural number k. Hence d(M-2-1h) + 1= d(M-i-1Ph) + 1=
d (d i -1) = i + 1 > 1. By Proposition 2.1 (1) and (3), we get
and
Hence
Hence we get the assertion.
COROLLARY 3.1. Let h•FZ•¨Z be a polynomial function such that Ph (t) 0Q[t] , and let i be an integer with 0_??_i_??_d(h). Then there exists h (t) E Z[t, t-1] such that Fh(t) = cbh(t)/(1 t)d with d = d(h) + 1 by Theorem 2.1. We put qh (t) >nEZ an (h)tn. Assume that h E P FF O. Then
PROOF. Since h e PFD°, by Remark 2.4 we get aj(h)=0 for any j<0. By the definition of d(ƒÓh), we get aj(h)=0 if j>d(ƒÓh). Hence we get the assertion by Theorem 3.1.
REMARK 3.1. If h E P.F'O and Ph(t) then d(cbh) > 0 by Remark 2.4. NOTATION 3.1. Let f•FZ•~Z•¨Z be a function in two variables. We put fl(x):=f(x, 0) and f2(y):=f(0, y).
DEFINITION 3.2. Let h(x) be a polynomial function in x. Then a function f•FZ•~Z•¨Z is called a polynomial function in two variables associated with h(x) if the following hold.
(1) f1(x)=h(x).
(2) There exists an integer N such that f2(n)=0 for every integer n with n_??_N.
NOTATION 3.2. Let h(x) be a polynomial function in x, and let f•FZ•~Z•¨Z be a polynomial function in two variables associated with h(x). Assume that h(x) E P2° and Ph (x) °Q{x} . Then by Theorem 2.1 and Remark 2.4 there exists a polynomial fi(t) E Z[t] such that
where aj•¸Z for every j. We put aj(f):=aj.
REMARK 3.2. Let h(x) be a polynomial function in x, and let f•FZ•~Z •¨ Z be a polynomial function in two variables associated with h(x). Assume that
h(x) E 'pj:'-° and Ph (x) 0Q[x] .
(1) By Notation 2.3 and Notation 3.2, q 51(t) _ ch(t).
(2) By Theorem 2.1 and (1) above, d(q f) = n(h) + d(h) + 1.
REMARK 3.3. Let h(x) be a polynomial function in x, and let f•FZ•~Z•¨Z L be a polynomial function in two variables associated with h(x). Assume that h(x) E P.°. We use Notation 3.1 and Notation 3.2. Then
(1) f1(O)=a0(f) and f1(1)=a1(f)+(d(h)+1)a0(f). (2) ad(~f)(f) 0.
DEFINITION 3.3. Let h(x) be a polynomial function in x, and let f•FZ•~Z•¨Z be a polynomial function in two variables associated with h(x). We use Notation 3.1.
(1) For every integer i with 0_??_i_??_d(h), the i-th sectional geometric genus gi(f) of f is defined by the following:
(2) For every integer i with 0_??_i_??_d(h), the i-th Ģ-genus Ģi(f) of f is defined by the following:
REMARK 3.4. (1) Let f•FZ•~Z•¨Z be a polynomial function in two vari ables associated with a polynomial function h(x). If i=1 and d(h)_??_1, then g1(f)=g(h) and ƒ¢1(f)=gƒ¢(h), that is, g1(f) is the sectional genus of h and
Ģ1(f) is the Ģ-genus of h (see Definition 2.2 and Remark 2.3 (2)).
(2) Let (X, L) be a polarized variety. We put f (x, y) :_ (-1)'hY(xL)
and h(x) := f (x, 0). Then we get gi(f)=gi(X, L) and Ģi(f)=Ģi(X, L) (see Definition 2.4 and Proposition 3.1).
THEOREM 3.2. Let h(x) be a polynomial function in x, and let f•FZ•~Z•¨Z be a polynomial function in two variables associated with h(x). We use Notation 3.2. Assume that h(x) E PJ''-0, Ph(t) # OQ[t}, and f(0, m)=f(1, m)=0 for every integer m with m_??_-1. Then
(1) For every integer i with 1_??_i_??_d(h)
If i=0, then 90(f)=Lik=O a (f).
(2) For every integer i with 1_??_i_??_d(h)
PROOF. First we note that ~bf(t) E Z[t] by Remark 2.4. We also note that
f2(0)=f(0,0)=h(0)=a0(f).
By Corollary 3.1, for 1_??_i_??_d(h)
we obtain
If i=0,
then
by Corollary
3.1 and Definition
3.3 we obtain
Next we prove (2) by induction on i. First we consider the case where i=1. Assume that d(ƒÓf)=0. Then ƒÓf(t)=a0(f). Hence by Remark 3.3 (1)
=0.
So we may assume that d(cb
f) > 1. Then by Remark 3.3 (1)
So we obtain the assertion for i=1.
Next we assume that the assertion is true for i=j_??_1.
We consider the case
where i=j+1_??_2.
Then by Definition 3.3 and assumptions
Therefore we get the assertion.
DEFINITION 3.4. Let h(x) be a polynomial function in x, and let f•FZ•~Z•¨ Z be a polynomial function in two variables associated with h(x). Here we use Notation 3.2. Then f is said to be h-semipositive if ai(f)_??_0 for every integer i.
COROLLARY 3.2. Let h(x) be a polynomial function in x, and let f•FZ•~Z•¨ Z be a polynomial function in two variables associated with h(x). Assume that h(x) E 'P.~~0, f is h-semipositive, and f(0, m)=f(1, m)=0 for every integer m with m_??_-1. Then we get the following:
(1) g(f) > Dj(f) > 0 for every integer i with 1_??_i_??_d(h).
(2) For every integer i with 1_??_i_??_d(h), gi(f)=0 if and only if Ģi(f)=0. (3) If gi(f)=0, then gi+1(f)=0 for every integer i with 1_??_i_??_d(h)-1.
PROOF. We use Notation 3.2.
(1) Here we note that, by assumption, for every integer i with 1_??_i_??_d(h)-1
On the other hand, since f is h-semipositive, by Theorem 3.2 we obtain Ģi(f)_??_0 for every integer i with 1_??_i_??_d(h). Hence if 1_??_i_??_d(h)-1, then gi(f)-Ģi(f)=
Ģi +1(f)_??_0. If 1_??_i= d(h), then by Theorem 3.2
Therefore we get the assertion of (1).
(2) By (1), if gi(f)=0, then Ģi(f)=0. Assume that Ģi(f)=0. Then by Theorem 3.2 (2), d(cb f) < i because ad(~ f) (f) > 0 and aj(f)_??_0 for every j by Remark 3.3 (2) and the h-semipositivity of f. Hence by Theorem 3.2 (1), we get gi(f)=0.
(3) If gi(f)=0, then by Theorem 3.2 (1) we get d(~ f) < i because ad(c f) (f) > 0. In particular d(q5 f) < i + 1. Therefore again by Theorem 3.2 (1) we obtain gi+1(f)=0.
As an application of Corollary 3.2, we get the following result concerned with polarized toric varieties. Here a pair (X, L) is called a polarized tonic variety if X is a toric projective variety and L is an ample line bundle on X.
THEOREM 3.3. Let (X, L) be a polarized tonic variety of dimension n. Then we get the following.
(1) gi(X, L)_??_Ģi(X, L)_??_0 for every integer i with 1_??_i_??_n.
(2) For every integer i with 1_??_i_??_n, gi(X, L)=0 if and only if Ģi(X, L)= 0.
(3) If gi(X, L)=0, then gi+1(X, L)=0 for every integer i with 1_??_i_??_n-1. PROOF. We put f (x, y) := (-i)h(xL) -y-vand h(x) := f (x, 0). By [12],
Proposition 2.4, Corollary 2.8, Corollary 2.9 and Corollary 2.14, we obtain
hi(mL)=0 for every pair of integers i and m with i_??_1
and m_??_0
and there exists
an integral convex polytope P in Rn such that dim P=n
and h0(mL)=i(P, m)
ger m' with m'_??_-1. Here we note that h0(mL)=0
for every integer m with
m<0.
Hence h E P.F~°, h(m) = h°(mL) = i(P, m) for m_??_0,
and h(m)=0 for
m<0.
So by Theorem 2.3, h(x) is h-semipositive. Therefore by Remark 3.4 (2)
and Corollary 3.2 we get the assertion.
COROLLARY
3.3. Let (X, L) be a polarized tonic variety of dimension n. For
every integer i with 0_??_i_??_n,
we obtain gi(X, L)_??_hi(Ox).
PROOF. If i=0,
then by Remark 2.6 (1) we get g0(X, L) = Ln > 1 =
h° (OX). Hence we may assume that i_??_1.
Then hi (OX)=0 by [12], Corollary 2.8.
So by Theorem 3.3 (1), we get the assertion.
THEOREM
3.4. Let (X, L) be a polarized tonic variety of dimension n.
(1) For every integer i with 0_??_i_??_n
(2) Assume that X is smooth and h° (KX + (n
i)L) > 0 for some integer i
with 0_??_i_??_n,
where KX is the canonical divisor of X. Then
(-1)(X,L)
ZxH
0 if i is odd,
2 if i is even.
PROOF. We use Notation 3.2. We put f (x, y) := (-1)Yh(xL)
--~and
h(x) :=
f (x, 0). Then h is a polynomial function and Ph(t)=x(tL).
Here we note that
a°(f) = h(0) = h°(OX) =1. First we prove (1).
(1.1) If d(~5f) < i + 1, then by Proposition 3.1 and Corollary 3.1 we have
xH (X, L)
H (h) = a° (f) =1. Hence
(1.2) If d(cb
f) > i + 1, then by Proposition 3.1 and Corollary 3.1 we get
=0
if i is odd,
2
if i is even
>(-1)i.
So we get the assertion of (1).
(2) Since X is smooth, L is very ample by [12] Proposition 2.4 (ii) and Corol lary 2.15. In particular h0(L)>0. Hence we obtain hn(-mL)=h0(KX+mL)>0
for every integer m with m_??_n-i because h0(KX+(n-i)L)>0. Here we note that Ph(m)=(-1)nhn(mL) for m<0 by the Kodaira vanishing theorem. Hence Ph(m)•‚0 for m_??_-(n-i). On the other hand h(m)=0 for every integer m with m<0. Therefore n(h) > -(n i) by the definition of n(h). Namely d(~bf) -1= d(h) + n(h) > i since d(cb f) = d(h) + n(h) + 1 and d(h) = n. Hence by (1.2) above, we get the assertion of (2).
4. Classification of finite partially ordered sets by their invariants DEFINITION 4.1. (1) Let P be a finite partially ordered set. We put
Then there exists a polynomial Q(t) such that deg Q(t)=d(P) and for every n•¸N
(See [17] 3.11). We set S~(P, t) := Q(t). This Ħ(P, t) is called the order polynomial of P.
(2) Let P be a finite partially ordered set and let Ħ(P, n) be the order poly nomial of P. We put
Then hp is a polynomial function, and we define a polynomial function in two
variables fp(x, y) associated with hp(x) as follows.
We call fp(x, y) the generalized polynomial function associated with P. We note
that d(P)=d(hp).
and
We note that g1(P)=gs(hp) and Ģ1(P)=gĢ(hp) (see also Definition 2.2 and Remark 2.3 (2)).
(4) Let P be a finite partially ordered set. For x, y•¸P, we say that x covers y if y<x and no element z•¸P satisfies y<z<x. The notation
means that x covers y. A finite partially ordered set is completely determined by
its cover relations.
DEFINITION
4.2. (1) Let P be a finite partially ordered set, and let C be a
subset of P. Then C is called a chain of P if any two elements of C are comparable.
(2) Let P be a finite partially ordered set, and let C be a chain of P. Then
we put l(C):=#(C)-1.
(3) Let P be a finite partially ordered set. Then we put
l(P):=max{l(C)|C
is a chain of P},
which is called the length of P.
(4) Let n be a natural number and let ‡”n be the set of all permutations of {1,.. . , n}. For ƒÐ•¸‡”n which satisfies ƒÐ(i)=ai for i=1,... , n, we use the following notation:
(5) Let P be a finite partially ordered set. We put P = {Xi,... , xd(p) } and A(P):={ƒÊ: P•¨{1, ... , d(P)}|
ƒÊ is a bijection such that ƒÊ(xi)<ƒÊ(xj) if xi<xj}.
(5.1) We fix an element ƒÊ•¸A(P). Then we put
By definition, we get L (P; p) C Ed(p). We call the set L(P; ƒÊ) the Jordan - Holder set of P with respect to ƒÊ (see [17], 3.12).
and
For a natural number n and a subset S C {1,... , n -1}, we put
Let ƒÊ, ƒÊ'•¸A(P) with ƒÊ•‚ƒÊ'. Then by [17] 3.12.1 Theorem, we obtain the following. For any subset S C {1,... , d(P) -1}, (Dd(p)(S) n £(P; µ)) _ j(Dd(p) (S) n £(P; ji')). In particular, for every integer i with 1_??_i_??_ d(P)-1 we get
So when we use results concerned with ƒÂ(ƒÎ) (for example, Proposition 4.1 below), we describe the Jordan-Holder set as L(P) instead of L(P; ƒÊ) for ƒÊ•¸ A(P).
PROPOSITION
4.1. Let P be a finite partially ordered set and let h(x) be the
following:
Then
PROOF. See [17] 4.5.14 Theorem.
REMARK 4.1. Let P be a finite partially ordered set, and let fP(x, y) be the generalized polynomial function associated with P.
(1) By Theorem 2.1 and Remark 2.4 there exists a polynomial ƒÓ(t)•¸Z[t] such that
where aj•¸Z for every j. Let aj(P):=aj. Then hp(t) E P.F~°, Php (t) °Q[t] , and f p(0, m) = f p(l, m) = 0 for every integer m with m•‚0. Hence by Theorem 3.2 we get the following:
(1.1) For every integer i with 1_??_i_??_d(P)
(1.2) If i=0, then g0(P) _ ~«o P a (P).
Moreover by the definition of the i-th Ģ-genus, we also get the following: (1.3) For every integer i with 1_??_i_??_d(P)-1,
(1.4) Since fp(x, y) satisfies the assumptions in Corollary 3.2, we get the following.
(A) gi(P)_??_Ģi(P)_??_0 for every integer i with 1_??_i_??_d(P).
(B) For every integer i with 1_??_i_??_d(P), gi(P)=0 if and only if Ģi (P)=0.
(C) If gi(P)=0, then gi+1(P)=0 for every integer i with 1_??_i_??_ d(P)-1.
(2) Since d(hp)=d(P), by Proposition 4.1 we get
Hence fp(x, y) is h-semipositive.
(3) If d(P)=1, then Ħ(P, t)=t and q c (t) = t. So by (1.1) above we get g1(P)=0 and Ģ1(P)=0.
THEOREM 4.1. Let P be a finite partially ordered set, and let i be an integer with 1_??_i_??_d(P). Then gi(P)=0 if and only if l(P)_??_d(P)-i.
PROOF. Here we use notation in Definition 4.2. (A) The case where d(P)=1.
Then l(P)_??_0=d(P)-1 always holds, and by Remark 4.1 (3), we get g1(P)=0.
(B.1) Assume that l(P)_??_d(P)-i. We put P = {pi,. .. , pd(P) }. By assump tion, there exists a subset {pt1, .. • , ptd(P}-i+1 } of P such that Pt1 < < ptd(P)-i+1
For every ƒÐ•¸L(P), there exist ƒÊ, ƒÎ•¸A(P) such that a _ ILolr-'. We put ir(p2) _: n2 E {1,. .. , d(P)} for every i with 1_??_i_??_d(P). Then nt1 < < ntd(P>-i+~. For every integer j with 1_??_j_??_d(P)-i, there exists u E {1,... , d(P)} such that nt~ < u < u + 1 <n1 and µ o l-1(u) < µ o 1t-i (u + 1) because IL(pt) < µ(pt~+1) Hence 5(a) < (d(P) -1) (d(P) i) = i -1 for every ƒÐ•¸L(P). By Proposition 4.1 and Remark 4.1 (1.1), we get gi(P)=0.
(B.2) Assume that l(P)<d(P)-i. First we prove the following claim. CLAIM 4.1. Let P be a finite partially ordered set with d(P)_??_2. Assume that l(P)=d(P)-k for some integer k with 1_??_h_??_d(P). Then there exists a member ƒÐ•¸L(P) such that ƒÂ(ƒÐ)=k-1.
PROOF. We prove this by induction on d(P).
(I) Assume that d(P)=2. Then we put P={p1, p2}. Let µ: {pi, p2} -~ {1, 2} (resp. ir: {pi, p2} --> {1, 2}) be a bijection such that ~t(p) = i (resp. 7r(pi) = 3 i). (I.1) Assume that p1 and p2 are comparable. Then we may assume that p1<p2. Then l(P)=1=d(P)-1, that is, k=1. On the other hand, since A(P)={ƒÊ}, we obtain
and
(I.2) Assume that p1 and p2 are incomparable. Then l(P)=0=d(P)-2, that is, k=2. On the other hand, since A(P)={ƒÎ, ƒÊ}, we get
and
By (I.1) and (I.2) we get the assertion for the case where d(P)=2.
(II) Assume that the assertion is true for the case where d(P)r_??_2.
Next
we consider the case where d(P)=r+1.
Let P={P1,. .. ,Pr+1}. We put
M(P):={the
maximal element of C
l := #M(P), and P:= P 1 M(P). Without loss of generality, we may assume that M(P) = {Pi,... , pi}. (Here we note that any two member of M(P) are incomparable.)
(11.1) If P=ƒÓ, then any two members of P are incomparable. In particular k=d(P)=r+1. Then there exist two bijections ,u, it : P -~ {1,. .. , r + 1 } such that 7r(p2) = r + 2 i and µ(p2) = i for every integer i with 1_??_i_??_r+1. In this case, it, µ E A(P) and µ o -i e £(P; pt). Moreover b(µ o it-1) = r = (r +1) -1= k -1. So we get the assertion in this case.
(11.2) If d(P)=1, then
P =
• , pr+1
pr+i
< p2
for
every
integer
i with
1_??_i_??_r}.
Namely k=d(P)-1=r.
Then there exist two bijections ,u, it : P -* {1,.. . , r + 1}
such that
and µ(p2) = r + 2 i for every integer i with 1_??_i_??_r+1.
In this case, it, t E A(P)
and t o it-i E £(P; ps). Moreover 8(p o it') = r -1= k -1. So we get the assertion
in this case.
(11.3) Assume that d(P) > 2. Then l(P) = l(P) -1= d(P)
k -1= r
k
(r + 1 l) (k l + 1) = d(P)
(k l + 1). By induction hypothesis there exists a
bijection map ~r : P -+ {1,... , r + 1 l} such that ~r e A(P) and 8(~r
o µ-1) = k l
for some µ E A(P). Here we put
and
Then it, t E A(P), ~0 11 i
~roµrl r+i
e £(P), and
Hence we get the assertion of Claim 4.1.
By this claim, if 1(P)<d(P)-i, then there exists a member ƒÐ•¸L(P) such that ƒÂ(ƒÐ)_??_i. Hence by Proposition 4.1 and Remark 4.1 (2), we have aj>0 for some j_??_i+1, and gi(P)>0 by Remark 4.1(1.1).
We obtain the assertion of Theorem 4.1.
By Theorem 4.1, Conjecture 1.1 is true. Namely we get the following result. COROLLARY 4.1. Let P be a finite partially ordered set. Then g1(P)=0 if and only if P is a totally ordered set.
PROOF. By Theorem 4.1, we can show that g1(P)=0 if and only if l(P)_??_ d(P)-1. On the other hand, P is a totally ordered set if and only if l(P)_??_d(P)-1. Therefore we get the assertion.
In [18], Yamamoto proved that g1(P)=0 if P is a totally ordered set. Next we give a classification of P with g2(P)=0 and g1(P)•‚0.
COROLLARY 4.2. Let P be a finite partially ordered set with d(P)_??_2. As sume that g2(P)=0 and g1(P)•‚0. Then P is one of the following types.
(A) d(P)_??_2
and
(Bi) d(P)_??_3 and
(Cj) d(P)_??_3
and
(Dk,l) d(P)_??_4
and
(Here in the case (Bi) (resp. (Cj)), i (resp. j) is any integer with 2_??_i_??_d(P)-1 (resp. 1_??_j_??_d(P)-2), and in the case (Dk,l), k and l are any pair of integers with 2_??_k+1<l_??_d(P)-1).
PROOF. By assumption and Theorem 4.1, we obtain 1(P)=d(P)-2. Hence P is one of the types in the statement of Corollary 4.2. Therefore we get the assertion.
REMARK 4.2. Assume that P is one of the types in Corollary 4.2. By Propo sition 4.1, we can calculate g1(P), Ģ1(P), g2(P) and Ģ2(P).
Assume that P is the type (A). We put
and
where t is an integer with 2_??_t_??_d(P)-1. Then
and
Hence by Proposition 4.1 we get a1(P)=1, a2(P)=d(P)-1,
and as(P)=0
for
every integer s with s_??_3.
Assume that P is the type (Bi). We put
and
where
t is an integer
with 2_??_t_??_. Then
and
Hence by Proposition 4.1 we get a1(P)=1, a2(P)=i-1, and as(P)=0 for every integer s with s_??_3.
Assume that P is the type (Cj). We put
and
where t is an integer with j+1_??_t_??_d(P)-1. Then
and
Hence by Proposition 4.1 we get al(P)=1,
a2(P)=d(P)-j-1,
and as(P)=0
for every integer s with s_??_3.
Assume that P is the type (Dk,l). We put
where
t is an integer
with 1_??_t_??_l-k.
Then
and
Hence by Proposition 4.1 we get a1(P)=1, a2(P)=l-k-1,
and as(P)=0
for
every integer s with s_??_3.
In each case, d(ƒÓfp)=2. So by Remark 4.1 (1.1) we get the following table.
THEOREM
4.2. Let P be a finite partially ordered set with d(P)_??_2. Assume
that g1(P)=1.
Then P is one of the following types.
(A) d(P)=2 and
(B2) d(P)_??_3
and
(Cd(p)-2) d(P)_??_3
and
(Dk,k+2) d(P)_??_4
and
(Here k is any integer with 1_??_k_??_d(P)-3.)
PROOF. Assume that g1(P)=1. Since g1(P)_??_ƒ¢1(P)_??_0 by Remark 4.1 (1.4) (A), we obtain ƒ¢1(P)=1 because of g1(P)•‚0 and Remark 4.1 (1.4) (B). Hence by Remark 4.1 (1.3) ƒ¢2(P)=g1(P)-ƒ¢1(P)=0. Then by Remark 4.1
(1.4) (B), we obtain g2(P)=0.
Since g2(P)=0 and g1(P)•‚0, P is one of the types (A), (Bi), (Cj) and (Dk,l) in Corollary 4.2. Since g1(P)=1, from the table in Remark 4.2, we get the assertion. E
