Singular lexicographic points on Hilbert schemes
aa r X i v : . [ m a t h . AG ] F e b SINGULAR LEXICOGRAPHIC POINTS ON HILBERT SCHEMES
RITVIK RAMKUMAR AND ALESSIO SAMMARTANO
Abstract.
We study the geometry of standard graded Hilbert schemes ofpolynomial rings and exterior algebras. We are motivated by a famous theo-rem of Reeves and Stillman for the Grothendieck Hilbert scheme, which statesthat the lexicographic point is smooth. By contrast, we show that, in standardgraded Hilbert schemes of polynomial rings and exterior algebras, the lexico-graphic point can be singular, and it can lie in multiple irreducible components.We settle questions of [I. Peeva, M. Stillman, Math. Ann. 339 (2007)] and of[D. Maclagan, G. Smith, Adv. Math. 223 (2010)]. Introduction
Hilbert schemes are fundamental parameter spaces in Algebraic Geometry. Themost classical example is the Grothendieck Hilbert scheme Hilb p p P n q [G61], whichparametrizes closed subschemes of P n with a fixed Hilbert polynomial p . However,it is often useful to consider the more general standard graded Hilbert scheme H h p R q , which parametrizes homogeneous ideals with a fixed Hilbert function h in a graded ring R ; see [HS04] for the general theory and several applications.Then Hilb p p P n q is a special case of H h p R q , when R is a polynomial ring and h isa sufficiently large truncation of p . Moreover, structural questions about variousloci in the Grothendieck Hilbert scheme are often approached via standard gradedHilbert schemes, see [CEVV09, DJNT17, E12] to name a few examples.To aid in the study of Hilbert schemes, it is beneficial to identify distinguishedpoints on them. Lexicographic ideals were introduced by Macaulay [M27] to clas-sify Hilbert functions and Hilbert polynomials. Several notable classes of Hilbertschemes, including Hilb p p P n q and H h p R q when R is a polynomial ring or an exte-rior algebra, possess a unique lexicographic point. Often, the lexicographic pointcan be used to obtain geometric and algebraic information on the Hilbert scheme,e.g. about connectedness [PS05, H66], irreducible components [R95], and syzy-gies [GMP11, MP12]. A fundamental result in this context was proved by Reevesand Stillman [RS97]: the lexicographic point on the Grothendieck Hilbert schemeHilb p p P n q is always smooth. The result is particularly strong, since Hilb p p P n q is smooth if n ď p p P n q , now known as the Reeves-Stillmancomponent. The theorem is useful in various situations, e.g. in questions of smooth-ness [SS20,R19b,S17], rationality [LR11], and in explicit constructions [G08,R19a].In some sense, the Reeves-Stillman theorem and Hartshorne’s connectedness theo-rem are the only general tools available in the complicated study of the geography Mathematics Subject Classification.
Primary: 14C05; Secondary: 13F20, 15A75.
Key words and phrases.
Standard graded Hilbert scheme; Reeves-Stillman component; lexico-graphic ideal; reducible scheme; exterior algebra. of Hilb p p P n q . A version of the Reeves-Stillman theorem holds for toric Hilbertschemes, where the role of the lexicographic point is played by the toric point [PS02].It is natural to ask whether the Reeves-Stillman theorem holds for standardgraded Hilbert schemes H h p R q , see [GMP11, p. 157]. The most prominent casesof interest are that of the polynomial ring R “ S , which provides the most naturalgeneralization of Hilb p p P n q , and the case of the exterior algebra R “ E , wherethe tangent space enjoys extra structure in terms of Gr¨obner flips [PS07]. Moregenerally, one would like to know whether the lexicographic point establishes acanonical component of H h p R q . In this paper we answer these questions negatively.For the polynomial ring, we prove the following result, which settles a questionof [MS10, p. 1610]. Theorem 1.
Let S “ k r x, y, z s and H “ H h p S q be the standard graded Hilbertscheme with Hilbert function h “ p , , , , , , , . . . q . Then H is the union of twoirreducible components of dimension 8, and the lexicographic point of H lies intheir intersection; in particular, the lexicographic point is singular.For the exterior algebra, we prove the following result, which settles a questionof [PS07, p. 546]. Theorem 2.
Let E “ Ź x e , . . . , e y and let H “ H h p E q be the standard gradedHilbert scheme with Hilbert function h “ p , , , q . Then H is the union of twoirreducible components of dimension 14 and 15, and the lexicographic point of H lies in their intersection; in particular, the lexicographic point is singular.2. Preliminaries
In this section we fix some notation and terminology. Throughout the paper k denotes an algebraically closed field with char p k q ‰
2. All rings and ideals inthis paper are N ´ graded k ´ vector spaces, and we will omit the word “graded”.We only consider algebras R that are finitely generated in degree 1, mainly thepolynomial ring S and the exterior algebra E , and their quotients.We denote by V d the graded component of degree d P Z of a vector space V . Weuse x f , . . . , f m y to denote the k -linear span of elements f , . . . , f m .The Hilbert function of an algebra R is the sequence p dim k R d : d P N q . Weonly consider Hilbert functions of algebras , therefore, by abuse of terminology, when I is an ideal, the phrase “Hilbert function of I ” refers to the Hilbert function of thealgebra presented by I .When R is a quotient of S , the h -vector p h , h , . . . , h s q of R is the vector ofcoefficients of the numerator of the Hilbert series of R . For fixed Krull dimension, h -vectors correspond bijectively to Hilbert functions. We have deg p R q “ ř si “ h i .If R is Cohen-Macaulay, the h -vector is the Hilbert function of a general Artinianreduction of R , and h i ě i .We denote by H h p R q the standard graded Hilbert scheme parametriz-ing ideals of R with Hilbert function h [HS04]. We denote by Hilb p p P n q the Grothendieck Hilbert scheme parametrizing closed subschemes of P n withHilbert polynomial p , equivalently, saturated ideals with Hilbert polynomial p .We fix the lexicographic order among monomials of S or E . A lexicographicideal is a monomial ideal L such that each L d is spanned by the dim k L d largestmonomials of degree d . Classical results of Macaulay and Kruskal-Katona state INGULAR LEXICOGRAPHIC POINTS ON HILBERT SCHEMES 3 that, for all h such that H h p R q ‰ H , there exists a unique lexicographic ideal L P H h p R q , for both R “ S and R “ E . See [GMP11] for more details.We let in lex p I q denote the initial ideal of an ideal I . There is a one-parameterfamily whose general fiber is I and whose special fiber is in lex p I q ; this phenomenonis known as Gr¨obner degeneration.We denote by Gr p r, V q , resp. Gr p r, n q , the Grassmannian variety parametrizing r -dimensional subspaces of V , resp. of k n . Recall that dim Gr p r, n q “ r p n ´ r q .A pencil of quadrics of R is a 2-dimensional subspace V Ď R .3. The standard graded Hilbert scheme of the polynomial ring
This section is devoted to the proof of Theorem 1. Let S “ k r x, y, z s be thepolynomial ring in 3 variables over k . We consider the standard graded Hilbertscheme H “ H h p S q , which parametrizes ideals of S with Hilbert function h “ p , , , , , , , . . . q in other words ideals I Ď S such that I “ I “ , dim k I “ , dim k I “ , dim k I d “ ˆ d ` ˙ ´ d ě . Since ideals I P H define subschemes V p I q Ď P of dimension 0 and degree 3,we consider Hilb p P q , and collect some basic facts in the next lemma. Lemma 3.1.
The Hilbert scheme
Hilb p P q is a smooth irreducible 6-fold. It isstratified by h -vectors into locally closed subschemes (3.1) Hilb p P q “ H p , q š H p , , q where H p , q is open and H p , , q is an irreducible divisor. The subscheme H p , , q is the locus of complete intersections of degrees t , u , while H p , q is the locus ofcodimension 2 ideals of minors of ˆ matrices of linear forms.Proof. The first statement is [F68, Theorem 2.4]. Every J P Hilb p P q is saturated,hence Cohen-Macaulay, of codimension 2 and degree 3. The only possible h -vectorsare p , q and p , , q , and this yields the stratification (3.1). The ideals J P H p , , q are of the form J “ p ℓ, c q where x ℓ y P Gr p , S q – P and x c y P Gr p , S { ℓS q – P .It follows that H p , , q is isomorphic to a P -bundle on a P , and therefore it isa closed irreducible 5-dimensional subscheme of Hilb p P q . The last statementfollows from the Hilbert-Burch theorem. (cid:3) We define three loci in the Hilbert scheme, according to the h -vector of thesaturation and the codimension of the quadratic part of an ideal. Definitions 3.2.
Let X be the closure in H of the locus X ˝ consisting of ideals I such that the ideal p I q has codimension 2.Let X be the locus of ideals I P H such that sat p I q has h -vector p , q and theideal p I q has codimension 1.Let Y be the locus of ideals I P H such that sat p I q has h -vector p , , q . Proposition 3.3.
The locus X Ď H is irreducible of dimension 8.Proof. We have a map X ˝ Ñ H p , q defined by I ÞÑ sat p I q . By Lemma 3.1, everyideal J P H p , q is generated by quadrics. Moreover, by upper semicontinuity,a general V P Gr p , J q generates an ideal of codimension 2. Let I P X ˝ with R. RAMKUMAR, A. SAMMARTANO sat p I q “ J . By definition, the ideal p I q has codimension 2, hence it is a completeintersection of 2 quadrics. Comparing the Hilbert functions of I and p I q , we deducethat I “ p I , J q , with I P Gr p , J q – P . Thus, the fiber over each J P H p , q is an open subset of a P . By Lemma 3.1 we conclude that X ˝ , and therefore also X , are irreducible of dimension 8. (cid:3) Lemma 3.4.
Every ideal J P H p , q is in the GL -orbit of one of the following (1) p xy, xz, yz q , (2) p x , xy, yz q , (3) p x , xy, xz ` y q , (4) p x , xy, y q .Proof. The subscheme V p J q Ď P has dimension 0 and degree 3, so it is sup-ported at 1, 2, or 3 points. More specifically, J “ X si “ q i where each q i is pri-mary supported at an isolated point, s ď
3, and ř si “ deg p q i q “
3. If s “ J “p x, y q X p x, z q X p y, z q “ p xy, xz, yz q . On the other hand, if the points are collinear,then J “ p ℓ, c q with ℓ P S , c P S , thus J has h -vector p , , q . If s “
2, then wemay assume deg p q q “ p q q “
2. Up to changes of coordinates, we have q “ p x, y q and either q “ p x, z q or q “ p x , z q . However, the former case cannotoccur, since J contains no linear form, therefore J “ p x, y q X p x , z q “ p x , xy, yz q .Finally, the cases with s “ (cid:3) Lemma 3.5.
We have X Ď X .Proof. The map X Ñ H p , q , defined by I ÞÑ sat p I q , stratifies X by the GL -orbitsof saturations. There are four strata X q , X q , X q , X q , corresponding to theorbits of Lemma 3.4, and it suffices to show that X i q Ď X for each i . Equivalently,it suffices to show that, for each of the four ideals J of Lemma 3.4 and every I P X with sat p I q “ J , we have I P X . We also point out that for all I P X , sincethe ideal p I q has codimension 1, I is spanned by two reducible quadrics with acommon factor, i.e., I is a pencil of reducible quadrics.Stratum X q : Let I P X with sat p I q “ J “ p xy, xz, yz q . There are 3 pencilsof reducible quadrics in J , namely x xy, xz y , x xy, yz y , and x xz, yz y , so we mayassume I “ x xy, xz y . Comparing the Hilbert functions of I and J we deduce that I “ ` xy, xz, yz p αy ` βz q , y z, y z , yz ˘ for some α, β P k . In order to show that X Ď X , it suffices to show that I is a limit of ideals of X ˝ when α, β ‰
0, since X is closed. A desired limit is I p t q “ ` xy ` ty p αy ` βz q , xz, y z, y z , yz ˘ ÝÑ p xz, xy, yz p αy ` βz q , y z, y z , yz q , in fact, we have I p t q P X ˝ because sat ` I p t q ˘ “ p xy ` tαy , xz, yz q is the ideal ofminors of the matrix ˆ x y ´ tαy y z ˙ and hence it belongs to H p , q by Lemma 3.1.Stratum X q : Let I P X with sat p I q “ J “ p x , xy, yz q . The pencils of reduciblequadrics in J are x x , xy y and x xy, yz y . If I “ x xy, yz y then we conclude that I “ ` xy, yz, x p αx ` βz q , x , x z, x z ˘ for some α, β P k . When α, β ‰
0, we have I P X since, as in the previous paragraph, it is a limit of ideals in X ˝ : I p t q “ ` yz ` tx p αx ` βz q , xy, x , x z, x z ˘ ÝÑ ` yz, xy, x p αx ` βz q , x , x z, x z ˘ . INGULAR LEXICOGRAPHIC POINTS ON HILBERT SCHEMES 5 If I “ x x , xy y then I “ ` x , xy, yz p αy ` βz q , y z, y z , yz ˘ , and the followinglimit for α, β ‰ I P X I p t q “ ` x ` t p αy ` βz q z, xy, y z, y z , yz ˘ ÝÑ ` x , xy, yz p αy ` βz q , y z, y z , yz ˘ . Stratum X q : Let I P X with sat p I q “ J “ p x , xy, xz ` y q . Since xz ` y is irreducible, the only pencil of reducible quadrics in J is x x , xy y . We get I “ ` x , xy, p xz ` y qp αy ` βz q ˘ ` p xz ` y q ` y, z q and the following limit for α, β ‰ I P X I p t q “ ` x ` t p αy ` βzy q , xy ´ t p αyz ` βz q ˘ ` p y ` xz q ` x, y, z ˘ Ñ ` x , xy, p y ` xz qp αy ` βz q ˘ ` p y ` xz q ` y, z ˘ . Stratum X q : Let I P X with sat p I q “ J “ p x , xy, y q . Up to changes of coor-dinates, we may assume I “ x x , xy y , then I “ p x , xy, y p αy ` βz q , y , y z, y z q ,and the following limit for α, β ‰ I P X I p t q “ ` x ` ty p αy ` βz q , xy, y , y z, y z ˘ ÝÑ ` x , xy, y p αy ` βz q , y , y z, y z ˘ . (cid:3) Proposition 3.6.
The locus Y Ď H is closed and irreducible of dimension 8.Proof. We have a map Y Ñ H p , , q defined by I ÞÑ sat p I q . An ideal J P H p , , q isof the form J “ p ℓ , c q with ℓ P S , c P S zp ℓ q . For every I P Y with sat p I q “ J ,by comparing Hilbert functions, we must have I “ p ℓ ℓ , ℓ ℓ , c , J q where x ℓ , ℓ y P Gr p , S q – P , x c y P Gr ˆ , J x ℓ ℓ , ℓ ℓ y S ˙ “ Gr p , q – P . Thus, the fiber over each J P H p , , q is a P -bundle over a P , and by Lemma 3.1we conclude that Y is irreducible of dimension 8. Note that Y Ď H is closed, sinceit is the preimage of the closed subset H p , , q Ď Hilb p P q . (cid:3) We are ready to state the main result of this section.
Theorem 1.
The standard graded Hilbert scheme H “ H h p S q is a union of twoirreducible components of dimension 8. The lexicographic point of H lies in theintersection of the two components, and is a singular point. Proof.
By Definitions 3.2 and Lemma 3.5 we have H “ X Y Y , and it follows fromPropositions 3.3 and 3.6 that X and Y are distinct irreducible components.The lexicographic ideal of H is L “ p x , xy, xz , y , y z q . Since sat p L q “ p x, y q ,we have L P Y . Now consider J P H defined by J “ p x , xy ` xz ´ y q ` xy p x, y, z q . We have J P X because sat p J q “ p x , xy ` xz ´ y , xy q is the ideal of minors of ˆ x y y ` z x y ˙ . We have L “ in lex p J q , so by Gr¨obner degeneration we obtain L P X as desired. (cid:3) R. RAMKUMAR, A. SAMMARTANO The standard graded Hilbert scheme of the exterior algebra
This section is devoted to the proof of Theorem 2. Let E “ Ź E be the exterioralgebra of a 5-dimensional vector space E “ x e , e , e , e , e y over k . We considerthe standard graded Hilbert scheme H “ H p , , , q p E q , which parametrizes idealsof E with Hilbert function p , , , q , in other words ideals I Ď E such that I “ I “ , dim k I “ , dim k I “ , I “ E , I “ E . We collect some simple facts about exterior algebras in the next lemma.
Lemma 4.1.
With notation as above, let ℓ i denote elements of E , let q, q i denoteelements of E , and let V be a subspace of E . (1) rank p q q “ ô q “ ℓ ^ ℓ for some ℓ i with dim k x ℓ , ℓ y “ . (2) rank p q q “ ô q “ ℓ ^ ℓ ` ℓ ^ ℓ for some ℓ i with dim k x ℓ , ℓ , ℓ , ℓ y “ . (3) x q , q y is a pencil of rank 2 quadrics ô x q , q y “ x ℓ ^ ℓ , ℓ ^ ℓ y for some ℓ i with dim k x ℓ , ℓ , ℓ y “ . (4) rank p q q ď ô q “ . (5) Let q “ ℓ ^ ℓ be of rank , then q P Ź V ô ℓ , ℓ P V . (6) Let q “ ℓ ^ ℓ ` ℓ ^ ℓ be of rank , then q P Ź V ô ℓ , ℓ , ℓ , ℓ P V . (7) Let q “ ℓ ^ ℓ ` ℓ ^ ℓ be of rank , then ℓ ^ q “ ô ℓ P x ℓ , ℓ , ℓ , ℓ y . We begin our analysis by distinguishing ideals based on pencils of rank 2 quadrics.
Lemma 4.2.
Let I Ď E be an ideal with Hilbert function p , , , q . If I contains nopencil of rank quadrics, then I Ď Ź V Ď E for some subspace V P Gr p , E q .Proof. Let I “ x q , q , q y . The ideal I contains rank 4 quadrics, hence we mayassume q “ e ^ e ` e ^ e . We claim that I Ď Ź V where V “ x e , e , e , e y .Assume by contradiction I Ę Ź V , for instance q R Ź V . Without loss ofgenerality e ^ e is in the support of q , and, up to scaling, q “ p e ` ℓ q ^ e ` q with q P Ź V and ℓ P x e , e , e y . If q is linearly independent from q then I contains 9 independent cubics e ^ q , e ^ q , e ^ q , e ^ q , e ^ q , e ^ q , e ^ q , e ^ q , e ^ q , contradicting dim k I “
8. Thus we may assume q “ q “ p e ` ℓ q ^ e .Let ℓ “ α e ` α e ` α e with α i P k . The change of coordinates e ÞÑ e ´ α e fixes q and allows to assume that α “
0. Likewise, and up to switching e and e , we may assume that ℓ “ αe where α “ α P k , so that q “ p e ` αe q ^ e .Now we claim that q P Ź V . First, observe that q ^ V “ Ź V Ď I and W ^ e Ď I , where W “ x e ^ e ` e ^ e , e ^ e , p e ` αe q ^ e , p e ` αe q ^ e y Ď ľ V. Since the two subspaces of I are disjoint and both have dimension 4, we deducethat I “ p Ź V q ‘ p W ^ e q . Now let q “ ℓ ^ e ` q with q P Ź V and ℓ P V .Using q we may assume ℓ P U “ x e , e , e y . Suppose that ℓ ‰
0. We have theinclusion q ^ U Ď I “ p Ź V q ‘ p W ^ e q , which, going modulo Ź V , becomes ℓ ^ e ^ U Ď W ^ e , equivalently, ℓ ^ U Ď W . In fact, we have ℓ ^ U Ď W X Ź U .However, it is easy to see that W X Ź U “ x e ^ p αe ´ e qy , contradicting thefact that dim k p ℓ ^ U q “
2. Therefore ℓ “ q P Ź V . INGULAR LEXICOGRAPHIC POINTS ON HILBERT SCHEMES 7
We have q ^ e P I X ` Ź V ^ e ˘ “ W ^ e , which yields q P W . Since q and q are independent, we may assume q P x e ^ e , p e ` αe q ^ e , p e ` αe q ^ e y . This implies that q “ p e ` αe q^ ℓ for some ℓ P U , since e ^ e “ p e ` αe q^ e .But then x q , q y is a pencil of rank 2 quadrics, contradicting the hypothesis. (cid:3) Inspired by Lemma 4.2, we define the following two loci in the Hilbert scheme.
Definitions 4.3.
Let X be the closure in H of the locus X ˝ consisting of ideals I “ p ℓ ^ ℓ , ℓ ^ ℓ , q q where ℓ i P E , q P E .Let Y be the closure in H of the locus Y ˝ consisting of ideals I “ p q , q , q , c q where q i P E , c P E are such that I Ď Ź V for some V P Gr p , E q , but I Ę Ź W for all W P Gr p , E q . Lemma 4.4.
Let I Ď E be an ideal with Hilbert function p , , , q containing apencil of rank quadrics. Then I lies in X Y Y .Proof. Up to changing coordinates, we may assume that I “ x e ^ e , e ^ e , q y ,q “ e ^ ℓ ` q , ℓ P t , e u and q P Ź V with V “ x e , e , e , e y . Observe thatthe following susbpaces of I are disjoint U “ x e ^ e , e ^ e y ^ E , U “ x e ^ q , e ^ q , e ^ q , e ^ q y . Since dim k U “ , dim k I “ , we deduce dim k U ď
3. Going modulo e thisyields dim k p q ^ V q ď
3, which in turn implies rank p q q ď . Write q “ ℓ ^ ℓ where ℓ , ℓ P V . We distinguish three cases.Case 1: Suppose first that ℓ “
0. Then q ‰
0, and consider the subspace W “ x e , e , e , ℓ , ℓ y Ď E . If dim k W “
5, then we may assume ℓ “ e , ℓ “ e .In this case the ideal p I q has Hilbert function p , , , q , so I “ p I q P X ˝ Ď X .If dim k W ď
4, then we may assume W Ď x e , e , e , e y . Thus ℓ , ℓ P x e , e , e y ,and we may assume ℓ P x e , e y . Changing coordinates in x e , e y , we may furtherassume ℓ “ e , so that q “ e ^ p αe ` βe q for some α, β P k with p α, β q ‰ p , q .The ideal p I q has Hilbert function p , , , q for every α, β , so I “ p I , c q for some c P E . If β ‰
0, then, using Lemma 4.1 (5), we see that I P Y ˝ Ď Y . Taking alimit β Ñ
0, we deduce that I P Y also when β “ ℓ “ e and q ‰
0. If e appears in q , then it followsthat dim k U ě
3, forcing I “ U ‘ U , so I “ p I q has Hilbert function p , , , q and I P X ˝ Ď X . If e does not appear in q , then q P Ź x e , e , e y . We get I ^ E ” q ^ E ” x e ^ e ^ e , q ^ e y p mod U q , so p I q has Hilbert function p , , , q , and I “ p I , c q for some c P E . Using Lemma 4.1 (5), (6), we verifythat I P Y ˝ Ď Y .Case 3: Suppose, finally, that q “
0, so ℓ “ e and I “ x e ^ e , e ^ e , e ^ e y .Since p I q has Hilbert function p , , , , q , it follows that I “ p I , c , c q with c , c P E . We may assume c P x e ^ e ^ e , e ^ e ^ e , e ^ e ^ e y , bypossibly using c to cancel e ^ e ^ e . Hence, c is the product of e and a(reducible) quadric in Ź x e , e , e y , and we may assume c “ e ^ e ^ e . Bythe same argument, we can choose c “ e ^ ℓ ^ ℓ with ℓ , ℓ P x e , e , e y . If x ℓ , ℓ y “ x e , e y then x c y “ x e ^ e ^ e y . Otherwise, we may assume ℓ “ e ` ℓ with ℓ , ℓ P x e , e y . Applying the change of coordinates e ÞÑ e ´ ℓ , and then a R. RAMKUMAR, A. SAMMARTANO change of coordinates in x e , e y , we fix I and c , while reducing ℓ ^ ℓ to e ^ e ,so that x c y “ x e ^ e ^ e y . To summarize, when q “ I to one of the two ideals K “ p e ^ e , e ^ e , e ^ e , e ^ e ^ e , e ^ e ^ e q ,L “ p e ^ e , e ^ e , e ^ e , e ^ e ^ e , e ^ e ^ e q . In order to conclude, it suffices to show that K P X and L P Y . They are initialideals K “ in lex p K q and L “ in lex p L q of the ideals K “ p e ^ e , e ^ e , e ^ e ` e ^ e q ,L “ p e ^ e , e ^ e , e ^ e ` e ^ e , e ^ e ^ e q , The conclusion follows by Gr¨obner degeneration, since K P X ˝ and L P Y ˝ . (cid:3) Lemma 4.5.
Let I Ď E be an ideal with Hilbert function p , , , q such that I Ď Ź V for some V P Gr p , E q . Then I lies in X Y Y .Proof. If I contains no rank 4 quadric, then it contains a pencil of rank 2 quadricsand the conclusion follows from Lemma 4.4. Without loss of generality we mayassume V “ x e , e , e , e y and I “ x e ^ e ` e ^ e , q , q y with q , q P Ź V .Observe that I ^ E “ ´ ľ V ¯ ‘ ´ I ^ e ¯ has dimension 7. It follows that I “ p I , c q for some c P E , and by Lemma 4.1 (6)we conclude I P Y ˝ Ď Y . (cid:3) Now we turn to parametrizing the loci X and Y . Proposition 4.6.
The locus X Ď H is irreducible and 14-dimensional.Proof. Consider the general member I P X , that is, an ideal I P X ˝ , which satisfies I “ p ℓ ^ ℓ , ℓ ^ ℓ , q q with Hilbert function p , , , q , such that ℓ i P E , q P E . We observe that ℓ ^ q “
0. In fact, if ℓ ^ q ‰
0, then by Lemma 4.1 (4), (7), rank p q q “ q P Ź V for some V P Gr p , E q with ℓ R V . However, as it follows fromthe discussion in the first paragraph of the proof of Lemma 4.4, this generates acontradiction.We parametrize I by choosing subspaces x ℓ y P Gr p , E q – P , (4.1) x ℓ , ℓ y P Gr ˆ , E x ℓ y ˙ – Gr p , q , (4.2) x q y P Z Ď Gr ´ , E x ℓ ^ ℓ , ℓ ^ ℓ y ¯ – P , (4.3)where Z Ď P is the locus of points x q y such that ℓ ^ q “ k ` x ℓ ^ ℓ , ℓ ^ ℓ , q y ^ E ˘ “ . Extending t ℓ , ℓ , ℓ u to a basis t ℓ , ℓ , ℓ , ℓ , ℓ u of E , a system of projective co-ordinates in this P is given by the coefficients λ i,j of the non-zero basis vectors INGULAR LEXICOGRAPHIC POINTS ON HILBERT SCHEMES 9 ℓ i ^ ℓ j . It is easy to see that the condition ℓ ^ q “ Q Ď P with equation λ , λ , ` λ , λ , ` λ , λ , “ , and Z Ď Q is the subset where (4.4) holds. We claim that condition (4.4) is openin Q , equivalently, that 8 is the largest possible dimension for the vector space in(4.4) as x q y P Q . If q “
0, then q is reducible, and, up to changing coordinates,the space x ℓ ^ ℓ , ℓ ^ ℓ , q y is generated by monomials; it is easy then to concludethat dim k ` x ℓ ^ ℓ , ℓ ^ ℓ , q y ^ E ˘ ď . If q ‰
0, then, since ℓ ^ q “
0, we have q “ ℓ ^ ℓ ` ℓ ^ ℓ for some ℓ , ℓ , ℓ P E by Lemma 4.1 (7), anddim k px ℓ ^ ℓ , ℓ ^ ℓ , q y ^ E q ` p ℓ ^ E qp ℓ ^ E q “ dim k p ℓ ^ ℓ ^ E q ` p ℓ ^ E qp ℓ ^ E q ď k p ℓ ^ E q “
6. We conclude that Z Ď Q is an open subset. It is also non-empty, since e.g. x ℓ ^ ℓ ` ℓ ^ ℓ y P Z ,therefore Z is irreducible of dimension 6.By definition, all ideals I P X ˝ arise in this way. Conversely, any such choices(4.1), (4.2), (4.3) determine an ideal in X ˝ , since, by construction, the resultingideal p ℓ ^ ℓ , ℓ ^ ℓ , q q has the correct Hilbert function p , , , q . Moreover, weclaim that each ideal I P X ˝ is obtained for a unique choice of subspaces (4.1),(4.2), (4.3). First, since I “ p I q , we observe that I contains a unique pencilof rank 2 quadrics: otherwise, up to changes of coordinates, we would have I “p ℓ ^ ℓ , ℓ ^ ℓ , ℓ ^ ℓ q or I “ p ℓ ^ ℓ , ℓ ^ ℓ , ℓ ^ ℓ q , and neither ideal has Hilbertfunction p , , , q . Given the uniqueness of the rank 2 pencil, the subspaces (4.1)and (4.2) are uniquely determined by I . In turn, it is obvious that the subspace(4.3) is uniquely determined by I .We have thus constructed an irreducible parametrization of X ˝ of dimension4 ` p ´ q ` “
14, so its closure X is also irreducible and 14-dimensional. (cid:3) Lemma 4.7.
Let U P Gr p , E q be such that dim k p U ^ E q ě . Then U Ď Ź W for some W P Gr p , E q if and only if U “ .Proof. One direction is obvious: if U Ď Ź W for some W P Gr p , E q , then U Ď Ź W “
0. Conversely, assume U Ę Ź W for all W P Gr p , E q , it sufficesto show that U contains a quadric q with rank p q q “
4, since then q ‰
0. Assumeby contradiction rank p q q ď q P U . Then any 2 quadrics in U share acommon factor, and without loss of generality U “ x e ^ e , e ^ e , q y . Since q R Ź x e , e , e y , we may assume q “ ℓ ^ e . If x ℓ y “ x e y , then dim k p U ^ E q “ x ℓ y ‰ x e y , then rank p e ^ ℓ ` ℓ ^ e q “ ℓ P x e , e y ,yielding a contradiction in either case. (cid:3) Proposition 4.8.
The locus Y Ď H is irreducible and 15-dimensional.Proof. Consider the general member I P Y , that is, an ideal I P Y ˝ , which satisfies I “ p q , q , q , c q with Hilbert function p , , , q , such that q i P E , c P E , I Ď Ź V for some V P Gr p , E q , and I Ę Ź W for any W P Gr p , E q . We observe that c is aminimal generator, since, writing E “ V ‘ x ℓ y , we have(4.5) dim k p I ^ E q ď dim k ´ ľ V ¯ ` dim k p I ^ ℓ q “ ` “ . We parametrize I by choosing subspaces V P Gr p , E q – P , (4.6) x q , q , q y P U Ď Gr ´ , ľ V ¯ – Gr p , q , (4.7) x c y P Gr ´ , E x q , q , q y ^ E ¯ – Gr p , q – P , (4.8)where U is the locus of points x q , q , q y P Gr p , q such that x q , q , q y Ę Ź W for every W P Gr p , E q , and such that dim k px q , q , q y ^ E q “
7. We claimthat U Ď Gr p , q is an open subset. We have already observed in (4.5) that 7 isthe largest dimension for x q , q , q y ^ E as x q , q , q y P Gr p , q , so the equationdim k px q , q , q y ^ E q “ x q , q , q y Ę Ź W for every W P Gr p , E q is also open. Finally, U ‰ H since e.g. x ℓ ^ ℓ , ℓ ^ ℓ , ℓ ^ ℓ y P U where V “ x ℓ , ℓ , ℓ , ℓ y . Therefore, U is irreducibleof dimension 3 p ´ q “ I P Y ˝ arise in this way. Conversely, any such choices(4.6), (4.7), (4.8) determine an ideal in Y ˝ , since the resulting ideal p q , q , q , c q has the correct Hilbert function p , , , q and the conditions of Y ˝ are satisfied.Moreover, each ideal I P Y ˝ is obtained for a unique choice of subspaces: it isobvious that the subspaces (4.7), (4.8) are uniquely determined by I , whereas for(4.6) this follows from the requirement that I Ę Ź W for any W P Gr p , E q .We have thus constructed an irreducible parametrization of Y ˝ of dimension4 ` ` “
15, so its closure Y is also irreducible and 15-dimensional. (cid:3) We are ready to state the main theorem of this section.
Theorem 2.
The standard graded Hilbert scheme H “ H p , , , q p E q is a union oftwo irreducible components of dimensions 14 and 15. The lexicographic point of H lies in the intersection of the two components, and is a singular point. Proof.
The classification of ideals in Lemmas 4.2, 4.4, 4.5 proves that H “ X Y Y ,whereas the parametrizations of Propositions 4.6, 4.8 show that X , Y are irreduciblesubschemes of the claimed dimensions. Obviously Y is not contained in X . Com-paring the minimal number of generators of the general member we deduce, byupper semicontinuity, that X is not contained in Y either. Thus, X and Y are twodistinct irreducible components of H .The lexicographic ideal of H is L “ p e ^ e , e ^ e , e ^ e , e ^ e ^ e , e ^ e ^ e q .We saw in the proof of Lemma 4.4 that L lies in Y . On the other hand, we have L “ in lex p L q where L “ ` e ^ e , e ^ e , e ^ e , e ^ e ^ e , e ^ e ^ p e ` e q ˘ . The change of coordinates e ÞÑ e ´ e shows that L is projectively equivalentto the ideal K P X of the proof of Lemma 4.4, so L P X , and by Gr¨obner degen-eration L P X as well. Thus, L belongs to the intersection of the two irreduciblecomponents, and in particular it is a singular point. (cid:3) INGULAR LEXICOGRAPHIC POINTS ON HILBERT SCHEMES 11
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E-mail address : [email protected] (Alessio Sammartano) Department of Mathematics, University of Notre Dame, NotreDame, IN 46556, USA E-mail address ::