A new compactification of the Drinfeld period domain over a finite field

Transcription

1 A new compactification of the Drinfeld period domain over a finite field Simon Schieder June 2009 Abstract We study a compactification of the Drinfeld period domain over a finite field which arises naturally in the context of Drinfeld moduli spaces. This compactification is in some sense dual to the compactification by projective space. It is normal but singular at the boundary. We construct a desingularization and obtain a smooth modular compactification of the Drinfeld period domain with a natural stratification, simple functorial description, and the boundary a divisor with normal crossings in the strongest sense. Dept. of Mathematics, ETH Zurich, 8092 Zurich, Switzerland, simonschieder@gmx.net

2 Introduction Let d be an integer and let V be a d-dimensional vector space over a finite field F q with q elements. Denote by S V the symmetric algebra on V over F q and by FracS V ) its field of fractions. Thus S V is non-canonically isomorphic to the polynomial ring over F q in d variables, but refraining from the choice of a basis for V will make the constructions and statements of the article more lucid and canonical. Let R V and RS V be the F q -subalgebras of FracS V ) defined as follows: [ ] R V := F q v v V {0} [ RS V := F q v, ] v v V {0} We make S V and RS V into graded rings by defining deg SV v) = deg RSV v) := and deg RSV v) :=. We make RV into a graded ring by defining deg RV v) :=. This definition will turn out to be more convenient for the remainder of the article. Associated to these rings we define the following schemes over F q : P V := Proj S V ) Q V := Proj R V ) Ω V := Spec RS V ) 0 ) Thus P V is non-canonically isomorphic to projective space P d F q. The scheme Ω V is equal to the open affine complement of the union of all F q -rational hyperplanes in P V, as well as the open affine subscheme of Q V obtained by inverting all homogeneous elements of degree in R V. It is an important example of a period domain over a finite field - the Drinfeld period domain. See for example Rapoport [8], Orlik [9] and Orlik-Rapoport [0]. The scheme Ω V is an analogue over F q of Drinfeld s period domain over a nonarchimedean local field F. The latter is defined as the complement in projective d )-space of the union of all F -rational hyperplanes, and makes sense only as a rigid analytic space, not as an algebraic variety. When F has equal positive characteristic, this period domain plays a central role in the analytic description of the moduli space of Drinfeld modules. See for example Deligne-Husemoller [] and Drinfeld [2]. The special case of Drinfeld modules of rank d with respect to the ring F q [t] and with a level structure of level t) leads naturally to the scheme Ω V defined above. The natural Satake, or Baily-Borel) compactification of this Drinfeld moduli space turns out to be essentially Q V, and not P V. As Q V is singular see section 4) we are thus led to the natural problem of constructing a good desingularization of Q V, which is the main goal of this article. 2

3 The details of the relation of Ω V, Q V and its desingularization with Drinfeld moduli spaces will not be discussed. We now describe the main results of the article. The study of the projective coordinate ring R V of Q V is carried out in sections 2 and 3. We determine a presentation of R V with generators and relations Corollary 2.5), prove that R V is integrally closed Proposition 2.2), and determine its Hilbert polynomial Propositions 2.6, 2.9, 2.0). Although R V is neither regular nor factorial Proposition 2.), we show that up to a Frobenius power, the ring R V is isomorphic to the symmetric algebra S V on the dual vector space V over F q see Theorem 3. for a precise statement). From this we again deduce that R V is integrally closed. In section 4 we study the projective variety Q V = ProjR V ). Applying the ring-theoretic results of sections 2 and 3, we deduce that Q V is a normal scheme, and compute its degree Corollary 4.). We construct a natural and well-behaved stratification of Q V where the strata are indexed by nonzero subspaces of the vector space V Theorem 4.6, Remark 4.7). This stratification is in some sense dual to the usual stratification of projective space P V Proposition 4.2), where the strata are indexed by nonzero quotients of V. We discuss the birational equivalence of Q V and P V Propositions 4.8, 4.9). Finally we prove that the singular locus of Q V is equal to the union of all strata of codimension at least 2 Theorem 4.0). Before turning our attention to the construction of a good desingularization of Q V in arbitrary dimension, we study the special cases where d = 2 and d = 3 in section 5. If d = 2, the schemes Q V and P V are isomorphic smooth projective curves, and in fact Q V = ProjR V ) is the q-uple embedding of the projective line P V Proposition 5.). If d = 3, the schemes Q V and P V are non-isomorphic surfaces. We prove that P V and Q V become isomorphic after blowing up both surfaces in every zero-dimensional stratum Theorem 5.2). Section 6 forms the heart of the article. Here we construct a desingularization B V of Q V in arbitrary dimension. Throughout the section we work exclusively with functors of points. Thus we begin by determining the functors represented by P V, Q V and Ω V Corollaries 6.3, 6.4). We define B V functorially and show that it is indeed representable by a projective variety over F q Proposition 6.5, Corollary 6.4). We then construct a stratification for B V where the strata are indexed by filtrations of the vector space V Theorem 6.9). The stratum corresponding to the trivial filtration is open and dense in B V and isomorphic to Ω V Proposition 6.7). The stratification enjoys several natural and beautiful geometric properties Proposition 6.6, Corollary 6.8, Remark 6., Corollary 6.5). We prove that B V is smooth Proposition 6.3, Corollary 6.4) and that the boundary B V Ω V is a divisor with normal crossings in the strongest sense Proposition 6.6). Finally, we construct projective morphisms to P V and Q V which are isomorphisms on Ω V. Therefore B V is a desingularization of Q V Corollary 6.7). 3

4 I would like to thank Prof. Richard Pink for initiating and guiding this project, for countless conversations on a wide range of mathematical topics, and more generally for his supervision and support during my time at ETH Zurich. 4

5 2 Structure of R V In this section we study the rings R V and RS V defined in the introduction. Our first goals are to find F q -bases for R V and RS V, and a presentation of R V with generators and relations. Not surprisingly, it is convenient to treat these questions simultaneously. Having found an F q -basis for R V, we deduce formulas for the Hilbert function and Hilbert polynomial of R V. At the end of the section we give an ad-hoc proof of the fact that R V is integrally closed. Let A V := F q [X v 0 v V ] denote the polynomial ring over F q in the indeterminates X v for all nonzero vectors v in V. Denote by κ V the degree-preserving surjection of graded rings κ V : A V R V X v v. Let J V A V be the homogeneous ideal generated by all homogeneous elements of the form for all 0 v V and α F q, and X v αx αv X v X v + X v X v + X v X v for all v, v, v V {0} such that v + v + v = 0. It is easy to see that the ideal J V is contained in the kernel of κ V. In Corollary 2.5 we will see that J V is actually equal to the kernel of κ V. Thus we will obtain a presentation of R V with generators and relations as R V = AV /J V. For the entire section, we choose an F q -basis e,..., e d of V. We call a monomial in A V reduced with respect to this basis if it is of the special form d i= X ei + i j= α ije j ) ri for some r i Z 0 and α ij F q. Let K V denote the F q -linear subspace of A V generated by all reduced monomials. Proposition 2. The F q -vector space A V is the sum of the linear subspaces J V and K V. Proof. We have to show that every monomial in A V lies in the sum J V + K V. By the definition of J V, we have the following two relations in the quotient A V /J V : 5

6 ) X αv = α X v for some α F q and 0 v V 2) X v X v = X v X v v X v X v v for some v v V {0} We will show that every monomial in A V can be transformed into an element of K V by using these relations finitely many times. For any nonzero vector v = d i= α ie i in V we define widthv) := max{i {,..., d} α i 0}. For any monomial f = n i= X v i in A V we define widthf) := n widthv i ). i= We carry out the proof by induction on the width w of the monomial f, starting with w = 0. By the usual convention that the empty sum is equal to 0 and the empty product is equal to, the case w = 0 implies f = and thus f K V. Now assume that the statement is true for all monomials of width less than w, and let f = n i= X v i have width w. Using relation ), it suffices to consider the case where every v i = d j= α ije j is monic in the sense that α ij = for j = widthv i ). If f is not already reduced, there exist indices j and k such that v j v k, but widthv j ) = widthv k ). Since v j and v k are monic, this implies that widthv j v k ) < widthv j ). Using relation 2) we can replace the factor X vj X vk by X vk X vj v k X vj X vj v k, so that ) ) f = X vi X vk X vj v k X vi X vj X vj v k. i j,k By construction, both summands have width less than w. Thus the induction hypothesis implies that both summands lie in J V + K V, hence also f. i j,k Proposition 2.2 The system S) of all elements of RS V of the following form is an F q - basis of RS V : ) d i ri e i + α ij e j i= j= for some r i Z, α ij F q, and α ij = 0 for all j if r i 0. For the second part of the proof of Proposition 2.2 we will need the following well-known lemma, whose proof we include for lack of a suitable reference. 6

7 Lemma 2.3 Let L be a field and let LT ) denote the field of rational functions in one indeterminate over L. Then the system of elements T a) n a L, n Z, and a = 0 if n 0 ) of LT ) is linearly independent over L. Proof. Assume there exists a non-trivial linear combination α a,n) T a) n = 0. a,n Since the subsystem T n n 0) of the system under consideration is already L-linearly independent, there must exist an element b L and an integer n < 0 such that α b,n) 0. Fix such an element b and let m denote the minimum of all integers n < 0 with α b,n) 0. We multiply the above linear combination with the least common multiple of all the denominators. Then every summand is divisible by T b) except for the summand corresponding to the index b, m). Thus we obtain an equation of the form T b) ft ) + α b,m) gt ) = 0, where f and g are nonzero polynomials in L[T ] and gb) 0. Substituting b for T yields the desired contradiction. Proof of Proposition 2.2. First we show that the system S) generates RS V over F q. Set i V i) := F q e j. j= Since the ideal J V is contained in the kernel of κ V and since κ V is surjective, Proposition 2. implies that κ V K V ) = R V. Unfolding the definitions, we see that the composition of κ V with the inclusion R V RS V maps the system of all reduced monomials of A V onto the subsystem of S) consisting of all elements ) d i ri e i + α ij e j with r i 0 for all i. i= j= Therefore, every element of RS V the form can be written as a linear combination of elements x of x = d i= el i i d i= e i + v i ) r i for some v i V i ), r i 0, l i 0. 7

8 After cancelling we can assume that whenever v i = 0, either r i = 0 or l i = 0. If x is not already an element of the system S), there exists an index i such that r i > 0 and l i > 0. Denote by n the largest such index i. We will now show that x can be written as a linear combination of elements of the same form, but with smaller value of n for each summand. Once this is shown, one can apply the same procedure to each of the summands. By performing at most d iteration steps, x can be written as a linear combination of elements of the above form with the additional property that for every index i, either r i = 0 or l i = 0, i.e., as a linear combination of elements of the system S). In order to find the desired linear combination of x, note that e ln n e n + v n ) rn = e n + v n v n ) ln e n + v n ) rn = l n k=0 ) ln e n + v n ) ln rn k v k k n. Now since v n V n ), the k-th power vn k on the right hand side can be written as a linear combination of products of basis vectors e i with i n. Thus multiplying this equation by the missing factors i n el i i i n e i + v i ) r i and subsequent cancelling yields the desired linear combination of x. Thus we have shown that the system S) generates RS V over F q. Now we come to the linear independence of the system S). We proceed by induction on d. The case d = is an immediate consequence of Lemma 2.3. We now assume that the statement holds for d ) indeterminates. Let T be a finite indexing set and assume that γ t b t = 0, where γ t F q and where b t := t T ) d i rit e i + α ijt e j i= is an element of the system S) under consideration. To simplify notation, denote v it := i j= α ijte j. In order to apply the induction hypothesis, we wish to arrange the summands of the finite sum t T γ tb t in a convenient way. As the index t runs over the indexing set T, the pair r dt, v dt ) takes values in the set Z F q e,..., e d ). More formally, we consider the map j= T Z F q e,..., e d ) t r dt, v dt ). 8

9 We partition T into the non-empty fibers of this map: For n Z and f F q e,..., e d ) define T n,f) := {t T r dt, v dt ) = n, f)}. Note that for n 0, the set T n,0) is the only potentially non-empty set among the T n,f). For notational purposes, define C n,f) := d ) γ t e i + v it ) r it. t T n,f) i= We calculate 0 = γ t b t t T = γ t e d + f) n n,f) t T n,f) = n,f) C n,f) e d + f) n. d ) e i + v it ) r it i= Since C n,f) lies in F q e,..., e d ) for all pairs n, f), we can apply Lemma 2.3 with ground field L = F q e,..., e d ) and coefficients C n,f). We conclude that C n,f) = 0 for all n, f). Note that every element C n,f) F q e,..., e d ) has precisely the form of an element of the system S) under consideration, only in one indeterminate less. Thus we can apply the induction hypothesis to each C n,f) and conclude that γ t = 0 for all t T. This finishes the proof of the linear independence of the system S). Corollary 2.4 The system consisting of all elements of R V F q -basis of R V : ) d i ri e i + α ij e j i= j= for some r i Z 0, α ij F q, and α ij = 0 for all j if r i = 0. of the following form is an Proof. We have already seen that J V kerκ V ). Since κ V : A V R V is surjective, Proposition 2. implies that R V = κ V K V ). Since κ V maps the system of all reduced monomials in A V to precisely the system under consideration, this shows that the system generates R V as an F q -vector space. The linear independence follows readily from Proposition 2.2. Corollary 2.5 The kernel of κ V is equal to the ideal J V. Thus we have found a presentation of R V by generators and relations as R V = AV /J V. Our choice of grading for R V in the introduction implies that this isomorphism is actually a degree-preserving isomorphism of graded F q -algebras. 9

10 Proof. It follows from Corollary 2.4 that the restriction κ V KV : K V R V is an isomorphism of F q -vector spaces, and that the F q -vector space A V decomposes as A V = J V K V. Thus kerκ V ) J V. The converse inclusion J V kerκ V ) has already been observed at the beginning of the section. Using the basis for R V of Corollary 2.4 we will now study the Hilbert function and the Hilbert polynomial of R V. We refer the reader to Matsumura [], section 3, and Bruns- Herzog [2], chapter 4, for background material. Recall from the introduction that we define deg RV ) :=, so that R v V becomes a Z 0 - graded ring. We denote the Hilbert function of R V by H d. This notation is well-chosen since the isomorphism class of R V only depends on the dimension d of the vector space V. Our first goal is to derive a recursion formula for the Hilbert function. Proposition 2.6 The value of the Hilbert function H d at an integer n Z 0 can be computed from the Hilbert function H d by the formula n H d n) = H d n) + q d H d k). Proof. Set V d ) := d i= F qe i and fix an integer n Z 0. Then H d n) is precisely the number of basis elements ) d i ri e i + α ij e j i= j= of degree n, where r i 0 and α ij F q as in Corollary 2.4. We count the number of such basis elements of degree n for fixed values of the exponent r d. Then the sum of these numbers will be equal to H d n). Any basis element of degree n decomposes uniquely into a product ) d rd b e d + α dj e j, j= where b is a basis element of R V d ) of degree n r d. Thus if r d = 0, the number of possibilities is equal to H d n). If r d < 0, the scalars α dj can be chosen arbitrarily in F q for all j =,..., d. Hence there are q d possibilities for the second factor and H d n r d ) possibilities for the first factor. Thus H d n) = H d n) + k=0 n n q d H d n r d ) = H d n) + q d H d k). r d = k=0 0

11 Remark 2.7 Using the formula of Proposition 2.6 above, one easily computes the Hilbert function of R V for small values of d. The following formulas hold for all integers n Z 0. H n) = H 2 n) = qn + q 3 H 3 n) = 2 ) n ) 2 q3 + q 2 + q n + q 3 + ) We now study the Hilbert polynomial of R V. We will need the following well-known lemma. Lemma 2.8 For integers m and n 0 denote by s m n) the m-th power sum s m n) := n k m. k= Then s m n) is a polynomial in n of degree m + ) and with leading coefficient m+. Lemma 2.8 can easily be proven directly by standard arguments using the difference function n) := s m n + ) s m n) = n m. Alternatively, see Conway-Guy [7] for an explicit formula for the coefficients of the polynomial s m n) in terms of Bernoulli numbers. We denote by P d the Hilbert polynomial of R V, i.e., the unique polynomial P d Q[n] such that P d n) = H d n) for all n 0. Due to an obstruction in cohomology, one cannot in general expect that the Hilbert function and the Hilbert polynomial agree for all n 0. This is however true for a polynomial ring, and, as we will now see, also for the ring R V. Proposition 2.9 i) P d n) = H d n) for all n 0. ii) The Hilbert polynomial of R V satisfies the same recursion formula as the Hilbert function, i.e., n P d n) = P d n) + q d P d k) for all integers n Z 0. Proof. We prove i) by induction on d. For the case d = see Remark 2.7 above. If the statement is true for d ), Proposition 2.6 above implies that k=0 n H d n) = P d n) + q d P d k) k=0

12 for all n 0. Now since P d n) is a polynomial in n, Lemma 2.8 implies that the right hand side of the equation is already a polynomial in n. This polynomial agrees with the Hilbert polynomial P d for all n 0, and therefore even for all n Z. Thus H d n) = P d n) for all n 0. The statement of ii) follows directly from i) and Proposition 2.6. Proposition 2.0 The Hilbert polynomial of R V has degree d ), and its leading coefficient is equal to d )! q d )d 2) 2. Proof. We proceed by induction on d. For the case d = we again refer to Remark 2.7 and part i) of Proposition 2.9 above. Now assume the statement is true for d ), i.e., for some a i Q and where a d 2 = d 2 P d n) = a i n i Using ii) of Proposition 2.9 above, we calculate i=0 d 2)! q d 2)d 3) 2. d 2 n d 2 P d n) = a i n i + q d a i k i = i=0 k=0 i=0 d 2 d 2 = a i n i + q d a i s i n ). i=0 i=0 Note that the polynomials s i n) and s i n ) Q[n] have the same degree and the same leading coefficient. Thus from Lemma 2.8 we see that P d n) has degree d ) and that its leading coefficient is equal to d qd a d 2 = d 2)d 3) d )! qd )+ 2 = d )! q d )d 2) 2. The following proposition collects some elementary properties of the ring R V. Proposition 2. i) The Krull dimension of R V is equal to d. ii) If d 2, the ring R V is not factorial. 2

13 iii) If d 2, the ring R V is not regular at the augmentation ideal m := i R V,i. Proof. The ring R V is a finitely generated F q -algebra and an integral domain. Thus the Krull dimension of R V is equal to the transcendence degree of its field of fractions FracR V ) over F q. Since FracR V ) is isomorphic to a function field over F q in d indeterminates, the transcendence degree of FracR V ) is equal to d. This proves i). Alternatively, one could appeal to Proposition 2.0, where it was proven that the degree of the Hilbert polynomial of R V equals d ). This again shows that the Krull dimension of R V is equal to d. We now prove ii). If d 2, we can choose pairwise linearly independent vectors v, v, v in V such that v + v + v = 0. The elements,, R v v v V are irreducible since they are homogeneous of degree. The equality v v + ) = v v v shows that divides the product, but it does not divide any of the two factors. Thus v v v is irreducible but not prime, so R v V cannot be factorial. To prove iii), we need to show that if d 2, the local ring R V,m at the maximal ideal m R V is not regular. Since R V is a finitely generated algebra over a field and an integral domain, every maximal ideal of R V has the same height. Therefore part i) of the proposition implies that the Krull dimension of R V ) m is equal to d. Because m is a maximal ideal, it suffices to show that dim Fq m/m 2 > d. The F q -algebra R V is generated in degree over R V,0 = F q. Thus dim Fq m/m 2 = H d ), where H d denotes the Hilbert function of R V as in Proposition 2.6 above. From the recursion formula in Proposition 2.6 we see that H d ) = H d ) + q d. By iterating and using that H ) =, we conclude that H d ) = d i=0 qi. If d 2, this number is greater than d. We finish the section with an ad-hoc proof of the fact that the domain R V closed. We will give a more conceptual proof in section 3 below. is integrally Proposition 2.2 The integral domain R V is integrally closed. Proof. It suffices to show that R V is integrally closed in the domain RS V, which is a localization of the regular ring S V and thus integrally closed in FracS V ). 3

14 Given an element b of the basis for RS V defined in Proposition 2.2, we define a pair of non-negative integers lb) Z 0 Z 0 as follows. If b is contained in R V, set lb) := 0, 0). If b is not contained in R V, there exists an integer i such that r i > 0. Then we set n := max{i r i > 0} and define Fix a total order on Z 0 Z 0 by defining lb) := n, r n ) Z 0 Z 0. x, y) > x, y ) : either x > x or if x = x, then y > y. For an arbitrary element x = i β ib i RS V for pairwise distinct basis elements b i RS V and scalars β i F q, define lx) := max lb i ). i Lemma 2.3 i) Let x, x RS V. Then lx + x ) maxlx), lx )). ii) Let x RS V and a R V. Then lax) lx). iii) Let x RS V R V and n Z >0. Then lx n ) > lx n ). Proof of Lemma 2.3. Part i) is clear from how we extended the definition of l from the special case of a basis element to an arbitrary element of RS V above. It follows from the algorithm used in the first part of the proof of Proposition 2.2 that statement ii) is true if a and x are basis elements. For arbitrary a = i α ia i and x = j β jb j with basis elements a i, b j and scalars α i, β j F q, we calculate lax) i) max la i b j ) max lb j ) = lx). i,j j The statement of iii) is clear if x is a basis element. If x = β i b i is arbitrary, fix an index k such that lx) = lb k ). Then we conclude that lx n ) = lb n k ) > lbn k ) = lx n ). Assume that x RS V R V is integral over R V. Fix an equation x n + a n x n a x + a 0 = 0 for some n > 0 and some a i R V. Then n ) l x n ) = l a i x i i) max l a i x i) ii) i n contradiction. i=0 max l x i) iii) = l iii) x n ) < l x n ), i n 4

15 3 Relating R V and S V It is the goal of this section to relate the rings R V and S V, where V denotes the dual vector space of V. We will show that R V and S V are isomorphic up to Frobenius in a sense made precise by Theorem 3. below. The majority of this section is devoted to the proof of Theorem 3.. As an application, we will use the theorem to give a more conceptual proof of the fact that R V is integrally closed. We begin by fixing some notation. For a linear form w V and a vector v V we define w, v := wv). Furthermore, we define the Frobenius homomorphisms F RV : R V R V, x x qd and F SV : S V S V, x x qd. Theorem 3. i) There exists a unique F q -algebra homomorphism ψ : S V R V such that w v. v V w,v = ii) There exists a unique F q -algebra homomorphism ϕ : R V S V v w V w,v = w. such that iii) iv) ϕ ψ = F SV ψ ϕ = F RV Proof of 3.i). We need to show that the formula for ψw) is F q -linear as w ranges over the vector space V. This is immediate once we write ψw) in the form ψw) = v V w,v = v = v V {0})/F q w, v v. 5

16 Proof of 3.ii). We first construct an F q -algebra homomorphism η : S V FracS V ) such that 0 v w. w V w,v = In order to show that such a homomorphism η exists, we need to verify that a) ηαv) = αηv) b) ηv + v ) = ηv) + ηv ) for all α F q and all 0 v, v V. To prove a), we calculate ηαv) = w V w,αv = w = w V w,v+v = w V w,v = α w = αqd w V w,v = w = αηv). Equation b) follows from equation a) if v and v are linearly dependent. If v and v are linearly independent, we have to prove that the formula w = w + w w V w,v = w V w,v = holds in FracS V ). After choosing an appropriate basis for V we can identify V with F d q and v, v with the standard basis elements, 0,..., 0), 0,, 0,..., 0) F d q. Denote by X,..., X d ) F d q the dual basis of the standard basis of F d q. Then we have to prove that the formula α F d q α +α 2 = d i= α ix i = α F d q α = d i= α ix i + α F d q α 2 = holds in the field of rational functions F q X,..., X d ). d i= α ix i We proceed by induction on the number of indeterminates d. For d = 2 we have to show that = +. αx + α)x 2 X + αx 2 αx + X 2 α F q α F q α F q Multiplying this equation by the factor X α F q X 2 αx ) we obtain the equivalent equation X 2 X ) = X + X 2. α α α Fq α 6 α Fq α 0

17 This equation in turn follows from the observation that α Fq α α = = α Fq α 0 We now assume that the statement is true for d ) indeterminates T,..., T d. We will deduce the statement for the d indeterminates X,..., X d by substituting the expressions X q i ) Xq d X i for Ti for all i =,..., d. Let l be a linear form over F q in two variables α, α 2. We calculate α. α F d q lα,α 2 )= d i= α it i = = = = α F d q lα,α 2 )= α F d q lα,α 2 )= α F d q lα,α 2 )= d i= α i X q i ) Xq d X i d ) q i= α ix i X q d d β F q d i= α ix i. i= α ix i + βx d d i= α ix i ) α F d q lα,α 2 )= By using this calculation for each of the three linear forms l : α, α 2 ) α + α 2, l : α, α 2 ) α, l : α, α 2 ) α 2, we deduce the statement for the d variables X,..., X d from the statement in the d ) variables T,..., T d. This finishes the proof of part b) above. We have therefore shown that the F q -algebra homomorphism η : S V FracS V ) is welldefined. Since ηv) 0 for all 0 v V, the map η extends uniquely from S V to RS V. By restricting this extension from RS V to its subring R V we obtain the desired map ϕ. Proof of 3.iii). We have to show that for any 0 w 0 V the equation w = w qd 0 v V w 0,v = w V w,v = 7

18 holds in S V. We choose a basis X,..., X d for V such that X d = w 0. By fixing the corresponding dual basis for V, we can identify V with F d q and the dual basis of X,..., X d with the standard basis of F d q. Then we need to show that the equation d α F d q α d = β F d q α,β = i= β i X i ) = X qd d holds in the field of rational functions F q X,..., X d ). Equivalently, we have to prove the equation ) d E) X d + β i X i α i X d ) = X qd d. α F d q β F d q i= We proceed by induction on the number of variables d. The case d = is clear. Since we will use the statement for two indeterminates in the induction step below, we treat the case d = 2 next. We calculate X 2 + β X + αx 2 )) = X q 2 X αx 2 ) q ) X 2 β F q α F q α F q = α F q X q 2 X 2 q X α F q α F q i=0 α q X q 2 X αx 2 = qx q 2 X 2 q X q i αx 2 ) i q = X 2 = X q 2, where for the last equality we used that { α i 0 if i q. = α F q if i = q. Here we define 0 0 :=, in accordance to our calculation above. i=0 X q i X i 2 α F q α i We now assume that equation E) holds for the d ) variables T,..., T d, i.e. ) d 2 T d + β i T i α i T d ) = T qd 2 d. α F d 2 q β F d 2 q i= 8

19 Given a scalar α d F q, we substitute the expression for T d, and the expression for T i for i =,..., d 2. X q d X d X d α d X d ) q X q i X i X d α d X d ) q Thus for each scalar α d F q we obtain an equation in the variables X,..., X d. We denote by E ) the sum of all of these equations. The right hand side of E ) is equal to α d F q X q d X d X d α d X d ) q ) q d 2. By the calculation for the case d = 2 above, this expression is equal to X q d )qd 2 = X qd d. The left hand side of E ) is equal to ) X q d X d X d α d X d ) q α F d q i= β F d 2 q d 2 )) + β i X q i α ix q d X i X d α d X d ) q q + α i X d X d α d X d ) = α F d q β F d 2 q d 2 X d + = α F d q i= β F d q d 2 ) q X d + β i X i α i X d ) i= ) ) β i X i α i X d ) X d α d X d ) q ) d X d + β i X i α i X d ). i= Thus we have deduced the desired formula in the d variables X,..., X d. Proof of 3.iv). Using iii) we see that ϕ ψ ϕ) F RV ) = FSV ϕ) ϕ F RV ) = 0. 9

20 Thus once we show that ϕ is injective, the assertion of iv) will follow from iii). To prove the injectivity of ϕ we first fix appropriate S V -algebra structures on the rings of interest for the proof. Consider R V and FracR V ) as S V -algebras via the map ψ : S V R V. Consider FracS V ) as an S V -algebra via the inclusion S V FracS V ). Finally, denote by SV F the ring S V endowed with the S V -algebra structure induced by the Frobenius map F SV. The following diagram of S V -algebras commutes because of the above choices of S V - algebra structures. R V ϕ S F V Frac S V R V SV Frac S V j S F V S V Frac S V Frac ψ i Frac R V Here we denote by Fracψ) the field homomorphism induced by ψ : S V R V. The map j is induced by ϕ : R V SV F and by the identity map of FracS V ). The map i is induced by the inclusion R V FracR V ) and by Fracψ). Since the diagram commutes, it suffices to show that j is injective. The fields FracR V ) and FracS V ) have the same transcendence degree d over F q, so that FracR V ) is algebraic over FracS V ) via the map Fracψ). Therefore FracR V ) is integral over the domain R V SV FracS V ) via the injection i. Thus the domain R V SV FracS V ) has to be a field as well, and j must be injective. Using Theorem 3. and the basis for RS V from Proposition 2.2, we will now give a more transparent proof of the fact that R V is integrally closed. Proof of Proposition 2.2. Let x FracR V ) be integral over R V. As in the proof given in section 2 we conclude that x RS V. Being injective, the maps ϕ and ψ of Proposition 3. extend to the fields of fractions of R V and S V. By abuse of notation we refer to these extensions as ϕ and ψ again. Since x is integral over R V, its image ϕx) FracS V ) is integral over the integrally closed domain S V. Thus ϕx) already lies in S V and its image ψϕx)) = x qd is an element of R V. 20

21 Thus it suffices to show that an element y RS V lies in the subring R V if and only if the Frobenius power y qd lies in R V. If y is an element of the basis for RS V constructed in Proposition 2.2, the Frobenius power y qd is again a basis element, and the statement about y follows directly from the shape of the basis under consideration. Since the Frobenius map is a homomorphism, we can reduce the general case to the case of a basis element by considering arbitrary linear combinations. 2

22 4 Structure of Q V We now study the projective variety Q V = ProjR V ). We begin with some basic properties of Q V which follow directly from the results of section 2. We then construct stratifications for Q V and for the projective space P V = ProjS V ) which are dual in a sense made precise by Proposition 4.2 and Proposition 4.6 below. Finally, we determine the singular locus of Q V and discuss the birational equivalence Q V Ω V P V. Corollary 4. i) The dimension of Q V is d ). ii) Q V is projectively normal. iii) The degree of Q V is equal to q d )d 2) 2. Proof. Part i) follows directly from Proposition 2., i). Part ii) is merely a repetition of Proposition 2.2. Part iii) follows from part i) together with Proposition 2.0. We now construct the aforementioned stratifications, beginning with the more familiar case of P V. A surjection of F q -vector spaces σ : V V 0 induces a degree-preserving surjection of graded F q -algebras S V S V with kernel v v kerσ)) S V. Thus we see that for any proper subspace V V, the scheme P V/V is the closed subscheme of P V corresponding to the homogeneous ideal v v V ) S V. The scheme Ω V/V is the locally closed subscheme of P V obtained by intersecting the closed subscheme P V/V with the open subscheme of P V on which the homogeneous element v V V v of S V does not vanish. Theorem 4.2 Stratification of P V ) The underlying set of the scheme P V is the disjoint union P V = Ω V/V. V V Proof. The disjointness is clear from the description of the strata Ω V/V as locally closed subschemes of P V. In order to see that every point of P V lies in one of the Ω V/V, let x P V and observe that the set V x := {v V vx) = 0} is a linear subspace of V. Then x lies in Ω V/Vx P V by definition of V x. Remark 4.3 The closure of a stratum Ω V/V P V is again a union of strata: Ω V/V = P V/V = V W V Ω V/W P V 22

23 We now proceed in a similar fashion to construct a stratification for the scheme Q V. Lemma 4.4 Let 0 V V be a nonzero linear subspace of V. Then there exists a degree-preserving surjection of graded F q -algebras ρ : R V R V such that v { v if v V {0}. 0 if v V V. Furthermore, the kernel of ρ is equal to the ideal v v V V ). Proof. We claim that the surjection π : A V := F q [X v 0 v V ] R V X v { v if v V {0}. 0 if v V V. has the kernel a := J V + X v v V V ). This suffices to prove the lemma because of the presentation R V = A V /J V of Proposition 2.5. The inclusion kerπ) a is immediate from the presentation R V = A V /J V. We now prove the converse inclusion. It is easy to see that all elements X v of a for some v V V and all elements of a of the form X v αx αv for some 0 v V and some α F q already lie in the kernel of π. Thus we only need to show that for three nonzero vectors v, v, v V with v + v + v = 0, the element f := X v X v + X v X v + X v X v a lies in the kernel of π as well. This is clear if at least two of the three vectors lie in V V. If not, then all three vectors must lie in V, so that f J V and therefore πf) = 0 as well. Corollary 4.5 Let 0 V V be a nonzero linear subspace of V. Then the scheme Q V is the closed subscheme of Q V corresponding to the homogeneous ideal v V V ) v R V. The scheme Ω V is the locally closed subscheme of Q V obtained by intersecting the closed subscheme Q V with the open subscheme of Q V on which the homogeneous element 0 v V of R v V does not vanish. For any three nonzero F q -vector spaces 0 V V V, the induced triangle of closed immersions Q V Q V Q V commutes. 23

24 Proof. The statements about Q V and Ω V are immediate from Lemma 4.4 above. The last statement follows from the commutativity of the triangle R V R V R V is the dis- Theorem 4.6 Stratification of Q V ) The underlying set of the scheme Q V joint union Q V = Ω V. 0 V V Proof. As in the proof of Proposition 4.2 above, the disjointness is a direct consequence of the description of the strata Ω V as locally closed subschemes of Q V. To show that any point x of Q V lies in one of the strata Ω V, we define the set V x := {v V {0} x) 0} {0}. v We claim that V x is a nonzero linear subspace of V. To see this, we need to show that if x) 0 and x) 0 for some v, w V, then also x) 0. This is immediate from v w v+w the equality v w = v + w v + ). w Thus x lies in the stratum Ω Vx Q V by definition of the subspace V x. Remark 4.7 The closure of a stratum Ω V Q V is again a union of strata: Ω V = Q V = 0 W V Ω W Q V Comparing the two stratifications 4.2 and 4.6 of P V and Q V, respectively, we observe that the sets of isomorphism classes of strata occurring in the two stratifications are identical. The disparity of the schemes P V and Q V is reflected in the functoriality of the indexing sets: The strata of P V are indexed by nonzero quotients of V, whereas the strata of Q V are indexed by nonzero subspaces of V. Since P V and Q V both contain the non-empty open stratum Ω V, they are birationally equivalent. Before studying this birational equivalence we introduce convenient open affine covers of P V and Q V. More precisely, for each stratum of P V and of Q V we construct the smallest 24

25 open affine neighborhood which is itself a union of strata. We denote the structure sheaves of P V and Q V by O PV and by O QV, respectively. We begin with the projective space P V. For a proper subspace V V, define U V/V := W V Ω V/W P V. We call U V/V the strata neighborhood of Ω V in P V. Note that U V/V is indeed the open affine subscheme of P V on which the homogeneous element v V V v of S V does not vanish. Therefore the affine coordinate ring of U V/V is equal to ) [ v O PV UV/V = Fq w v V, w V V ]. We proceed analogously for the scheme Q V. For a nonzero subspace 0 V V, define V V := V W Ω W Q V. We call V V the strata neighborhood of Ω V in Q V. It is indeed the open affine subscheme of Q V on which the homogeneous element 0 v V of R v V does not vanish. Thus the affine coordinate ring of V V is equal to [ v ] O QV V V ) = F q v V, w V {0}. w Proposition 4.8 The morphism P V Ω V Q V identifying the open stratum Ω V in P V with the open stratum Ω V in Q V can be extended uniquely to the union of all strata of P V of codimension. This extension map collapses each -codimensional stratum Ω V/V of P V to the corresponding 0-dimensional closed stratum Ω V of Q V. Proof. Let V V be a -dimensional proper subspace of V. Recall that ) [ v ] O PV UV/V = Fq v V, w V V [ w v ] O QV V V ) = F q v V, w V {0} w [ v ] RS V ) 0 = O QV Ω V ) = O PV Ω V ) = F q v V, w V {0} w We prove the existence of an extension to the strata neighborhood U V/V P V by providing a dotted arrow which makes the following diagram of affine coordinate rings commute: 25

26 ) O PV UV/V O QV V V ) RS V ) 0 Since the dimension of V is equal to, the quotient v lies in F w q for any two non-zero vectors v, w V. Thus [ v ] O QV V V ) = F q v V, w V V. w Hence we can define the desired dotted arrow to be the inclusion t : O QV V V ) O PV UV/V ). We now show that the -codimensional stratum Ω V/V of P V is mapped onto the 0- dimensional closed stratum Ω V of Q V under the corresponding map of affine schemes. The ideal I of the closed subscheme Ω V/V U V/V is equal to v ) I = v V, w V V O PV UV/V ). w Therefore, its inverse image t I) = v w v V, w V V ) O QV V V ) is precisely the ideal of the closed subscheme Ω V V V, as desired. Since Q V is separated and P V is reduced, any extension of the morphism P V Ω V Q V to an open subset containing Ω V P V is unique. Thus all the extensions obtained by varying the -dimensional proper subspace V V can be glued together. The following proposition can be proved in exactly the same fashion: Proposition 4.9 The morphism Q V Ω V P V identifying the open stratum Ω V in Q V with the open stratum Ω V in P V can be extended uniquely to the union of all strata of Q V of codimension. This extension map collapses each -codimensional stratum Ω V of Q V to the corresponding 0-dimensional closed stratum Ω V/V of P V. Theorem 4.0 The singular locus of Q V consists of all strata of codimension at least 2: Q sing V = dimv/v 2) Ω V. 26

27 Proof. Let 0 V V be a non-zero linear subspace of V. Choose a linear subspace V V such that V = V V. Our goal is to construct a morphism of schemes over F q θ : V V Ω V Spec R V which restricts to an isomorphism of a neighborhood of the closed subscheme Ω V in V V onto a neighborhood of the closed subscheme Ω V {0} in Ω V Spec R V. Here we denote by {0} the vertex of the affine cone Spec R V. This will link the problem of determining whether the points of Ω V are singular in Q V to the singularity of the ring R V, which has already been treated in Proposition 2.. We now construct the morphism θ. Fix a nonzero vector v 0 V. From the presentation of R V in Proposition 2.5 it is easy to see that there exists a unique F q -algebra homomorphism [ ] R V = F q v V {0} v such that F q [ v w v v 0 v. ] v V, w V {0} = O QV V V ) It is clear that this map is injective. Furthermore, there exists a natural inclusion [ v ] RS V ) 0 = F q v, w V {0} O QV V V ). w After identifying the rings RS V ) 0 and R V with their images in O QV V V ) under these injections, we observe that their intersection is trivial in the sense that RS V ) 0 R V = F q O QV V V ). Thus the induced F q -algebra homomorphism ε : RS V ) 0 Fq R V O QV V V ) is injective as well. To simplify notation we identify the ring RS V ) 0 Fq R V with its image [ v ] [ ] F q v, w V v 0 {0} w v v V {0} in O QV V V ). The homomorphism ε induces the desired map of affine schemes θ : V V Ω V Spec R V. We now show that θ satisfies the property stated at the beginning of the proof. The closed subscheme Ω V V V corresponds to the ideal v ) I := v V, w V V w 27

28 in O QV V V ). The inverse image v ε I) = w in the ring RS V ) 0 Fq R V = F q [ v w ) v V, w V {0} ] [ v, w V v 0 {0} v ] v V {0} is precisely the ideal corresponding to the closed subscheme Ω V {0} Ω V Spec R V. Therefore θ restricts to an isomorphism of the closed subscheme Ω V V V onto the closed subscheme Ω V {0} Ω V Spec R V. We claim that the injective F q -algebra homomorphism ε s : ) RS V ) 0 Fq R V O s Q V V V )) s obtained by localizing ε with respect to the homogeneous element s := ) v 0 + v 0 RS v v V ) 0 Fq R V 0 v V 0 v V is surjective and thus an isomorphism. To see this, we have to show that for nonzero vectors u, u V and u V u, the element of O u +u QV V V )) s already lies in the localization RSV ) 0 Fq R V ). This follows from the equality s ) u v u + u 0 + v 0 = u u u u v 0 u ) v since the element 0 + v u 0 is invertible in the localization ) RS u V ) 0 Fq R V. s Since θ : V V Ω V Spec R V is the map of affine schemes corresponding to the ring homomorphism ε : RS V ) 0 Fq R V O QV V V ), the fact that the localized homomorphism ε s is an isomorphism implies that θ is an isomorphism away from the zero-loci of s in V V and Ω V Spec R V. Therefore, our next goal is to show that the zero-locus of s in V V is disjoint from Ω V, and that the zero-locus of s in Ω V Spec R V is disjoint from Ω V {0}. We need to show that given nonzero vectors v V and v V, the homoge- ) v neous element 0 + v v 0 vanishes nowhere on Ω v V V V and Ω V {0} Ω V Spec R V. For a point x Ω V V V or x Ω V {0} Ω V Spec R V, we calculate v 0 v + v 0 v by the definition of Ω V, as desired. ) x) = v 0 v x)

29 We have now shown that the map θ induces an isomorphism of a neighborhood of Ω V in V V with a neighborhood of Ω V {0} in Ω V Spec R V, and that this isomorphism is an extension of the isomorphism Ω V = ΩV {0}. This implies that θ yields a bijection Q sing V Ω V ) = ΩV Spec R V ) sing Ω V {0}). Since Ω V is smooth, we note that Ω V Spec R V ) sing = Ω V Spec R V ) sing. It was already shown in Proposition 2. that the vertex {0} of Spec R V is a singular point if and only if dim V 2. Thus the subset Ω V Q V consists of only non-singular points if dim V/V, and of only singular points if dim V/V 2. 29

30 5 Low-dimensional examples In this section we study the special cases d = 2 and d = 3. If d = 2, the varieties P V and Q V are isomorphic curves. If d = 3, they are non-isomorphic surfaces. In this case, we will prove that the blow-up of P V in every closed stratum is isomorphic to the blow-up of Q V in every closed stratum. In particular, the blow-up of Q V or of P V ) is a desingularization of Q V in this special case. We first assume that d = 2. Then the curve Q V is non-singular according to Theorem 4.0. In fact, Propositions 4.8 and 4.9 imply that the curves Q V and P V are even isomorphic in this case. Knowing that Q V is a smooth curve, this of course also follows from the general fact that up to isomorphism there is a unique smooth projective curve in every birational equivalence class. In a similar vein, we now show that the map ϕ of Theorem 3. yields an isomorphism of Q V with the q-uple embedding of P V. Proposition 5. Let d = 2. Then the map ϕ : R V S V of Theorem 3. induces an isomorphism of graded F q -algebras R V = S q) V := i 0 S V, qi. Thus the projective curve Q V P V = ProjS V ). = ProjR V ) is the q-uple embedding of the projective line Proof. The degree of the homogeneous element ϕ ) S v V is equal to q for any nonzero vector v V. Thus ϕ factors through a degree-preserving map R V S q) V. From Theorem 3. we already know that this map is injective. We now prove that it is also surjective. Since the map is degree-preserving, F q -linear and injective, it suffices to show that the graded F q -algebras R V and S q) V have the same Hilbert function. The Hilbert function H 2 of R V was determined to be H 2 n) = qn + in Remark 2.7 above. Since dimv ) = d = 2, this is precisely the Hilbert function of S q) V. For the remainder of the section let d = 3. Thus P V and Q V are surfaces. According to Theorem 4.0 the singular locus of Q V is the union of all 0-dimensional strata. A 0-dimensional stratum of Q V or P V consists of precisely one closed point. In this situation, Propositions 4.8 states that the morphism P V Ω V Q V can be extended to the union of all strata of P V of codimension by collapsing each -dimensional stratum of P V to the corresponding 0-dimensional stratum of Q V. Proposition 4.9 provides the analogous statement for the morphism Q V Ω V P V. We will now show that by 30

31 blowing up both surfaces in all 0-dimensional strata one obtains isomorphic objects. For background material on blowing up we refer the reader to Hartshorne [5], II.7, Eisenbud [3], 5.2., and Eisenbud-Harris [4], IV.2. Denote by P V P V and by Q V Q V the blow-ups of P V and of Q V, respectively, in all the 0-dimensional strata. Thus the open stratum P V Ω V Q V is also a dense open subset of P V and Q V. Theorem 5.2 Let d = 3. Then the identity morphism P V Ω V = Ω V Q V extends uniquely to an isomorphism of the blow-ups P V = QV. In particular, Q V is a desingularization of Q V. Proof. Let 0 V V 2 V be a complete flag of V. We will construct open affine subschemes A V,V 2 of P V and B V,V 2 of QV which contain Ω V and are isomorphic via a map extending the identity on Ω V. We will show that if 0 V V 2 V ranges over all complete flags of V, the open sets A V,V 2 cover P V and the open sets B V,V 2 cover Q V. Since the blow-ups P V and Q V are separable and reduced, this implies that the identity map P V Ω V = Ω V Q V extends uniquely to an isomorphism P V = QV. We begin with the construction of the affine open subscheme A V,V 2 of P V. Denote by Ũ V/V2 the inverse image of [ the strata neighborhood U V/V2 under the projection map P V P V. Denote by A := F v ] q w v V, w V V2 the affine coordinate ring of UV/V2 and by I := ) v w v V2, w V V 2 the ideal in A corresponding to the closed point ΩV/V2 of U V/V2. Thus Ũ V/V2 = Proj Bl I A), where Bl I A := A I I 2... denotes the blow-up algebra of A with respect to I. Inverting the homogeneous element f := v V 2 V w V V 2 v w Ik Bl I A of degree k := #V 2 V ) #V V 2 ) in Bl I A yields the desired open affine subscheme A V,V 2 of P [ ] V, with affine coordinate ring equal to Bl I A). This affine coordinate ring is equal to F q [ v w = F q [ v w ] [ v v V, w V V 2 w w v ] v V 2, w V V, f 0 ] v V 2, v V 2 V, w, w V V 2 where the last equality is a consequence of the fact that dim V 2 = 2: 3

32 The second ring is clearly contained in the first ring. For the converse inclusion, one sees easily that it suffices to show that for given vectors v V and w V V 2, the quotient v is an element of the second ring. To see this, note that the vector space V decomposes w as V = V 2 F q w since dim V 2 = 2 = dim V. Hence there exist elements v 2 V 2 and α F q such that v = v 2 + αw. Thus v = v 2 w w + α is indeed an element of the second ring. We now proceed analogously with the construction of the affine open subscheme B V,V 2 of Q V. Denote by ṼV the inverse image of the strata neighborhood V V under the projection map Q [ V Q V. Denote by B := F v q w v V, w V {0} ] the affine coordinate ring of V V and by J := ) v w v V, w V V the ideal in B corresponding to the closed point Ω V of V V. Thus Ṽ V = Proj Bl J B). Inverting the homogeneous element g := v V {0} w V 2 V v w J l Bl J B of degree l := #V {0}) #V 2 V ) in Bl J B yields the desired open affine subscheme B V,V 2 of Q [ ] V, with affine coordinate ring equal to Bl J B). Using that any two nonzero vectors of the -dimensional vector space V only differ by multiplication with a scalar in F q, we see that this affine coordinate ring is equal to [ v ] [ ] v F q v V, w V {0} w w w v v, v V {0}, w V V, w V 2 V [ v ] [ ] w = F q v V, w V V w w w V V, w V 2 V [ v ] = F q v V 2, w V V. w g 0 Thus the open subscheme A V,V 2 P V is isomorphic to the open subscheme B V,V 2 Q V via a map extending the identity on Ω V. It remains to show that the constructed open subsets of the blow-ups are indeed coverings. We begin with the blow-up P V. For a fixed 2-dimensional subspace V 2 of V, we prove that if V ranges over all -dimensional subspaces of V 2, the open sets A V,V 2 cover Ũ V/V2. Let p be a homogeneous prime ideal of Bl I A such that for every -dimensional subspace V of V 2, the homogeneous element f V := v w Ik Bl I A v V 2 V w V V 2 32

33 of degree k = #V 2 V ) #V V 2 ) lies in p. We have to show that p contains the augmentation ideal I I 2... of Bl I A. Denote by V p the set consisting of all vectors v V 2 for which there exists a vector w V V 2 such that the homogeneous element v w I Bl I A of degree lies in p. The set V p is in fact a subspace of V. We have to show that V p is equal to V 2. Choose a -dimensional subspace V of V 2. Since f V lies in p, there exists a nonzero vector v V 2 V such that v V p. Set V := F q v. Then since f V lies in p, there exists a nonzero vector v V 2 V such that v V p. By construction, the vectors v and v are linearly independent. Thus V p is equal to V 2. We now prove the analogous result for the blow-up Q V. For a fixed -dimensional subspace V of V, we prove that if V 2 ranges over all 2-dimensional subspaces of V containing V, the open sets B V,V 2 cover ṼV. Let q be a homogeneous prime ideal of Bl J B such that for every 2-dimensional subspace V 2 of V containing V, the homogeneous element g V2 := v w Il Bl J B v V {0} w V 2 V of degree l = #V {0}) #V 2 V ) lies in q. We have to show that q contains the augmentation ideal J J 2... of Bl J B. Choose a generator v of V. Denote by V q the set consisting of 0 V and of all nonzero vectors w V such that the homogeneous element v w J Bl J B of degree does not lie in q. It is easy to see that the set V q is in fact a subspace of V. Since V q contains V, the dimension of V q is at least. We have to show that the dimension of V q is in fact equal to one. Choose any 2-dimensional subspace V 2 of V containing V. Then since g V2 lies in q, there exists a nonzero vector w V 2 V which does not lie in V q. Hence the dimension of V q is at most 2. If it was equal to 2, the fact that the homogeneous element g Vq lies in q yields a contradiction to the fact that q is a prime ideal. Thus the dimension of V q is equal to, as desired. 33