Rational normal curves and Hadamard products
Enrico Carlini, Maria Virginia Catalisano, Giuseppe Favacchio, Elena Guardo
RRATIONAL NORMAL CURVES AND HADAMARD PRODUCTS
ENRICO CARLINI, MARIA VIRGINIA CATALISANO, GIUSEPPE FAVACCHIO, AND ELENA GUARDO
Abstract.
Given r > n general hyperplanes in P n , a star configuration of points is the setof all the n -wise intersection of them. We introduce contact star configurations , which arestar configurations where all the hyperplanes are osculating to the same rational normal curve.In this paper we find a relation between this construction and Hadamard products of linearvarieties. Moreover, we study the union of contact star configurations on a same conic in P ,we prove that the union of two contact star configurations has a special h -vector and, in somecases, this is a complete intersection. Introduction
We say that the hyperplanes in a set
L = { (cid:96) , . . . , (cid:96) r } ⊆ P n , r ≥ n , meet properly if (cid:96) i ∩ ⋯ ∩ (cid:96) i n is a point for any choice of n different indices and n + (cid:96) i ∩ ⋯ ∩ (cid:96) i n by P i ,...,i n .Let L = { (cid:96) , . . . , (cid:96) r } ⊆ P n be a set of r ≥ n hyperplanes meeting properly. The set of points S (L) = ⋃ ≤ i < ... < i n ≤ r P i ,...,i n ⊆ P n . is called a star configuration of points in P n defined by L . These configurations of points, and their generalizations, have been intensively studied fortheir algebraic and geometrical properties, see [2, 4, 7, 16, 25] for a partial list of papers thathave contributed to our understanding them.Set S = C [ x , . . . , x n ] = C [ P n ] , where C could be replaced by any algebraically closed field ofcharacteristic zero. We recall that the Hilbert function of a set of points X ⊆ P n is the numericalfunction H X ∶ Z ≥ → Z ≥ defined by H X ( t ) = dim S t − dim ( I X ) t , where I X is the ideal defining X, and the h -vector of a set of points X ⊆ P is the first differenceof the Hilbert function of X , that is h X ( t ) = H X ( t ) − H X ( t − ) , where we set H X (− ) = . A star configuration S (L) defined by a set of r hyperplanes consists of ( rn ) points, and its h -vector is generic, see for instance [16, Theorem 2.6], that means h S (L) = ( , . . . , ( n − + in − ) , . . . , ( r − n − )) . Mathematics Subject Classification.
Key words and phrases.
Complete intersection, Hadamard product, star configuration, Gorenstein.Last updated: February 11, 2021. a r X i v : . [ m a t h . AG ] F e b E. CARLINI, M. V. CATALISANO, G. FAVACCHIO, AND E. GUARDO
Indeed, the ideal defining S (L) is minimally generated in degree r − r − L .We now construct star configurations starting from a rational normal curve γ of P n . We callthem contact star configurations on γ , we will not mention γ if it is clear from the context. Definition 1.1.
Let P , . . . , P r ⊆ P n be distinct points on a rational normal curve γ of P n .Denoted by L = { (cid:96) , . . . , (cid:96) r } the set of osculating hyperplanes to γ at P , . . . , P r respectively. Wesay that S (L) is a contact star configuration on γ .Note that, since γ is a rational normal curve, the hyperplanes in L always meet properly.The first motivation to introduce these configurations come from Hadamard products. Weshow in Section 2 that the so called Hadamard star configurations are indeed contact star con-figurations, see Theorem 2.1. This result will give an easy way to explicitly construct exampleswhich only make use of rational points, see in Remark 3.3.The second motivation is related to their h -vector. A single contact star configuration has ageneric h -vector, as any other star configuration. But the behavior of a union of two or more ofthem deserves further investigation. The homological invariants of a set of points union of starconfigurations have been studied for instance in [2, 26, 27]. In the known cases, that requiresome restrictive assumptions, the h -vector of such a union is always general.We will mostly focus on P , therefore the contact star configurations are defined by takinglines tangent to an irreducible conic. The study of properties of families of lines tangent to aplanar conic is classical in algebraic geometry, see for instance the Cremona’s book [10].We prove, in Section 3, that the union of two contact star configurations in P , defined by of r and s lines is a complete intersection of type ( r − , s ) if either s = r − s = r , see Theorem3.1. We also show that, in these cases the curve of degree s can be chosen irreducible.Moreover, in Section 4, we prove that the union of contact star configurations in P defined r and s lines has the same h -vector of two fat points of multiplicities r − s −
1, see Theorem4.3. We believe that this correspondence with the h -vector of certain scheme of fat points alsooccurs for a union of three and four contact stars, see Conjecture 4.8. We prove it in some cases,see Theorem 4.5.In Section 5, we apply Theorem 3.1 to the study of a recurring topic in classical projective ge-ometry: polygons circumscribed around an irreducible conic in P , see Proposition 5.1, Corollary5.2 and Proposition 5.3.Section 6 contains concluding remarks and conjectures for further work.We will make use of standard tools from the linkage theory, see [24] for an overview in the topicand [14, 15, 16, 21, 23] for a partial list of papers which use liaison to study zero-dimensionalprojective and mutiprojective schemes. A well know result, see [22, Corollary 5.2.19], relatesthe h -vectors of two arithmetically Cohen-Macaulay schemes in P n with the same codimension,that are linked by an arithmetically Gorenstein. In particular, if X, Y are two disjoint set ofreduced points in P and X ∪ Y is a complete intersection of type ( a, b ) , then the followingformula connects the h -vectors of X , Y and X ∪ Y (1.1) h X ∪ Y ( t ) = h X ( t ) + h Y ( a + b − − t ) , for any integer t. ATIONAL NORMAL CURVES AND HADAMARD PRODUCTS 3
Since the h -vector of a complete intersection is well known, from the formula above, the knowl-edge of the h -vector of X allows to compute the one of Y . Acknowledgments.
Favacchio and Guardo have been supported by Universit`a degli Studi diCatania, piano della ricerca PIACERI 2020/22 linea intervento 2. All the authors have beensupported by the National Group for Algebraic and Geometrical Structures and their Appli-cations (GNSAGA-INdAM). This work was partially supported by MIUR grant Dipartimentidi Eccellenza 2018-2022 (E11G18000350001). Our results were inspired by calculations withCoCoA [1]. 2.
Hadamard products
In this section we show that Hadamard star configurations are contact star configurations.Hadamard products of linear spaces have been recently subject of study for many interestingproperties, see for instance [3, 6, 7]. We briefly recall some general fact about Hadamard productsof linear spaces. Let P = [ a ∶ ⋯ ∶ a n ] and Q = [ b ∶ ⋯ ∶ b n ] be two points in P n . If for some i wehave both a i ≠ b i ≠
0, then we say that the
Hadamard product of P and Q , denoted P ⋆ Q ,is defined and we set P ⋆ Q = [ a b ∶ ⋯ ∶ a n b n ] ∈ P n . Given two varieties X and Y in P n , with respect to the Zariski topology, the Hadamardproduct of X and Y , denoted X ⋆ Y , is given by X ⋆ Y = { P ⋆ Q ∣ P ∈ X, Q ∈ Y, and P ⋆ Q is defined } ⊆ P n . In particular, for a variety X in P n and a positive integer r ≥
2, the r -th Hadamard power of X is X ⋆ r = X ⋆( r − ) ⋆ X, where we define X ⋆ = X .When we compute the Hadamard product of X and Y it often crucial to ensure some conditionof generality on X and Y , this is encoded by not having too much zero coordinates in theirpoints. For this purpose, we let ∆ i be the set of points of P n which have at most i + X is afinite set of points in P n , then the r -th square-free Hadamard product of X is X ⋆ r = { P ⋆ ⋯ ⋆ P r ∣ P , . . . , P r ∈ X distinct points } . From [3, Theorem 4.7] it is known that X ⋆ n is a star configuration of ( mn ) points of P n , where X ⊆ P n is a set of m > n points on a line (cid:96) such that (cid:96) ∩ ∆ n − = ∅ .Let V be a linear space in P n , for a positive integer r we consider the subscheme V ○ r = { P ⋆ r ∣ P ∈ V } ⊆ P n , called the r -th coordinate-wise power of V . Properties of these schemes have been studied in[12]. In the following theorem, we investigate the case where V is a line. Theorem 2.1.
Let (cid:96) be a line in P n such that (cid:96) ∩ ∆ n − = ∅ . Then (i) (cid:96) ○ n = { P ⋆ n ∣ P ∈ (cid:96) } is a rational normal curve; E. CARLINI, M. V. CATALISANO, G. FAVACCHIO, AND E. GUARDO (ii) let P ∈ (cid:96) , then the linear subspace of dimension d osculating to (cid:96) ○ n at P ⋆ n (i.e., itsintersection with (cid:96) ○ n is supported only on P ) is P ⋆( n − d ) ⋆ (cid:96) ⋆ d . In particular, the osculatinghyperplane to (cid:96) ○ n at P ⋆ n is P ⋆ (cid:96) ⋆( n − ) ; (iii) for each set of n distinct points on (cid:96) , P , . . . , P n ∈ (cid:96), we have P ⋆ P ⋆ ⋯ ⋆ P n = P ⋆ (cid:96) ⋆( n − ) ∩ ⋯ ∩ P n ⋆ (cid:96) ⋆( n − ) . Proof.
The degree of (cid:96) ○ n is n from Corollary 2.8 in [12]. Take a parametrization of the line (cid:96) , say P ab = [ L ( a, b ) ∶ L ( a, b ) ∶ . . . ∶ L n ( a, b )] ∈ (cid:96) where the L i ( a, b ) are linear forms in the variables a, b. Note that, since (cid:96) ∩ ∆ n − is empty, the forms L i ( a, b ) are pairwise not proportional.(i) The curve (cid:96) ○ n is parametrized by P ⋆ nab = [ L ( a, b ) n ∶ L ( a, b ) n ∶ . . . ∶ L n ( a, b ) n ] ∈ (cid:96) ○ n , wherethe components of P ⋆ nab are a basis for the forms of degree n in a, b since the L i ( a, b ) arepairwise not proportional. Hence, (cid:96) ○ n is a rational normal curve of P n .(ii) We will prove item (ii) by induction on d . Let d =
1. Now let P + tQ be a point of (cid:96) ,( t ∈ C ), thus the tangent line to (cid:96) ○ n at P ⋆ n islim t → ⟨ P ⋆ n , ( P + tQ ) ⋆ n ⟩ = lim t → ⟨ P ⋆ n , P ⋆ n + ntP ⋆( n − ) ⋆ Q + ⋅ ⋅ ⋅ + t n Q ⋆ n ⟩= ⟨ P ⋆ n , P ⋆( n − ) ⋆ Q ⟩ = P ⋆( n − ) ⋆ (cid:96). Assume d >
1. By the induction hypothesis, the linear space of dimension d − (cid:96) ○ n at P ⋆ n is P ⋆( n − d + ) ⋆ (cid:96) ⋆( d − ) . Let Q ≠ P be a point on (cid:96) . We have P ⋆( n − d + ) ⋆ (cid:96) ⋆( d − ) = { P ⋆( n − d + ) ⋆ ( a P + b Q ) ⋆ . . . ⋆ ( a d − P + b d − Q ) ∣ a i , b i ∈ C }= ⟨ P ⋆ n , P ⋆( n − ) ⋆ Q, P ⋆( n − ) ⋆ Q ⋆ , . . . , P ⋆( n − d + ) ⋆ Q ⋆( d − ) ⟩ . Now let again P + tQ be a point of (cid:96) , ( t ∈ C ). The linear space of dimension d osculatingto (cid:96) ○ n at P ⋆ n can be obtained by computing the following limitlim t → ⟨ P ⋆( n − d + ) ⋆ (cid:96) ⋆( d − ) , ( P + tQ ) ⋆ n ⟩ , and this limit, by an easy computation and the equality above, becomeslim t → ⟨ P ⋆ n , P ⋆( n − ) ⋆ Q, P ⋆( n − ) ⋆ Q ⋆ , . . . , P ⋆( n − d + ) ⋆ Q ⋆( d − ) , ( P + tQ ) ⋆ n ⟩= lim t → ⟨ P ⋆ n , P ⋆( n − ) ⋆ Q, . . . , P ⋆( n − d + ) ⋆ Q ⋆( d − ) , P ⋆ n + ntP ⋆( n − ) ⋆ Q + ⋅ ⋅ ⋅ + t n Q ⋆ n ⟩= lim t → ⟨ P ⋆ n , P ⋆( n − ) ⋆ Q, . . . , P ⋆( n − d + ) ⋆ Q ⋆( d − ) , ( nd ) t d P ⋆( n − d ) ⋆ Q ⋆ d + ⋅ ⋅ ⋅ + t n Q ⋆ n ⟩= ⟨ P ⋆ n , P ⋆( n − ) ⋆ Q, . . . , P ⋆( n − d + ) ⋆ Q ⋆( d − ) , P ⋆( n − d ) ⋆ Q ⋆ d ⟩= P ⋆( n − d ) ⋆ (cid:96) ⋆ d . (iii) It follows from (ii) and a well known property of rational normal curves in P n . (cid:3) In the next remark we give more details for n = Remark 2.2.
Consider a line (cid:96) in P and the respective conic (cid:96) ○ , let C [ x, y, z ] be the coordinatering of P . We have the following facts. ATIONAL NORMAL CURVES AND HADAMARD PRODUCTS 5 (i) Say (cid:96) defined by the equation αx + βy − z =
0, where α, β ≠
0. Then, from Theorem 2.1(i)we have that (cid:96) ○ is a conic. Precisely, one can check that (cid:96) ○ ∶ ( α x + β y − z ) − α β xy = . (ii) From Theorem 2.1(ii), for each P ∈ (cid:96) the line P ⋆ (cid:96) is tangent to (cid:96) ○ at P ⋆ P .(iii) From Theorem 2.1(iii), for any P, Q ∈ (cid:96) , P ≠ Q , the two tangent lines to (cid:96) ○ through P ⋆ Q are P ⋆ (cid:96) and Q ⋆ (cid:96) . Note that this allows us to find an explicit Hadamard decompositionof any point in the plane P = (cid:96) ⋆ (cid:96) . In fact, let A ∈ P , let a and b be the tangent linesto the conic (cid:96) ○ through A , and let P ⋆ P = a ∩ (cid:96) ○ , Q ⋆ Q = b ∩ (cid:96) ○ , then A = P ⋆ Q .(iv) For any P, Q ∈ (cid:96) , P ≠ Q , the line through P ⋆ P and Q ⋆ Q is the polar line of the point P ⋆ Q with respect to (cid:96) ○ .(v) The condition (cid:96) ∩ ∆ = ∅ ensures that (cid:96) meets the lines x = y = z = P x , P y and P z , respectively. Note that, from the definition ofHadamard product, P x ∗ (cid:96) is the line x = P y ∗ (cid:96) is the line y = P z ∗ (cid:96) is z =
0. Then, the conic (cid:96) ○ is tangent to the coordinate axes in P x ∗ P x , P y ∗ P y and P z ∗ P z .3. Complete intersections union of two contact star configurations in P In this section γ is an irreducible conic in P , and we set S = C [ x, y, z ] = C [ P ] . The mainresult of this section is the following Theorem. We postpone its proof until page 7, after thedevelopment of some special case. Theorem 3.1.
Let X = S (L) and Y = S (M) be two contact star configurations in P on thesame conic, where L = { (cid:96) , . . . , (cid:96) r } and M = { m , . . . , m s } are two sets of distinct lines. Then (a) if s = r − , then general form in ( I X ∪ Y ) r − is irreducible; (b) if s = r − , then X ∪ Y is a complete intersection of type ( r − , r − ) ; (c) if s = r , then the general form in ( I X ∪ Y ) r is irreducible; (d) if s = r , then ( I X ∪ Z ) r − = ( I X ∪ Y ) r − where Z denotes a set of r − collinear points in Y ; (e) if s = r , then X ∪ Y is a complete intersection of type ( r − , r ) . Remark 3.2.
Theorem 3.1 (e) in particular claims that the twelve points union of two contactstar configurations on the same conic, X = S ( (cid:96) , (cid:96) , (cid:96) , (cid:96) ) and Y = S ( m , m , m , m ) lie on acubic. This case is pictured in Figure 1. Remark 3.3.
Combining Theorem 3.1 and Theorem 2.1 we are able to explicitly give thecoordinates of a (not trivial) complete intersection of rational points in P of type ( a, b ) , whereeither b = a or b = a −
1. First, we need to fix a rational line (cid:96) in P . Then, we take a + b + (cid:96) divided in two sets X and Y containing a + b points, respectively.Then X ⋆ ∪ Y ⋆ is the complete intersection we are looking for. For instance, let (cid:96) be defined bythe linear form x + y − z and a = b =
3. We pick on (cid:96) the following points X = {[ ∶ ∶ ] , [ ∶ ∶ ] , [ ∶ ∶ ] , [ ∶ ∶ ]} and Y = {[ ∶ − ∶ ] , [ ∶ − ∶ − ] , [ ∶ − ∶ − ]} . Then, the set of 9 points X ⋆ ∪ Y ⋆ = {[ ∶ ∶ ] , [ ∶ ∶ ] , [ ∶ ∶ ] , [ ∶ ∶ ] , [ ∶ ∶ ] , [ ∶ ∶ ]} ∪{[ ∶ ∶ ] , [ ∶ ∶ ] , [ ∶ ∶ ]} E. CARLINI, M. V. CATALISANO, G. FAVACCHIO, AND E. GUARDO
Figure 1.
A cubic through twelve points union of two contact star configura-tions.from Theorem 3.1 (b), is a complete intersection of type ( , ) . Remark 3.4.
Note that if r = s =
2, then X ∪ Y consists of two points and the statements (c),(d), (e) in Theorem 3.1 are trivially true. The statements (a), (b) in the case r = , s = X ∪ Y consists of four points in linear general position. The first interestingcase occurs when r = s = r = s = Lemma 3.5.
Let X = S ( (cid:96) , (cid:96) , (cid:96) ) and Y = S ( m , m , m ) be contact star configurations on asame conic. Then (i) the general cubic through X ∪ Y is irreducible; (ii) a conic containing 5 points of X ∪ Y contains X ∪ Y ; (iii) X ∪ Y is a complete intersection of a conic and a cubic.Proof. Since the linear system of the cubics through X ∪ Y is not composite with a pencil and itdoesn’t have a common component, then by Bertini’s Theorem, see for instance [20, Section 5]and [19, Corollary 10.9], the generic cubic of the system is irreducible. In order to complete theproof, let C , C ′ be two cubics union of lines through X ∪ Y , (see Figure 2B). The intersection C ∩ C ′ consists of 9 points and, by Brianchon’s Theorem, see [8, pp. 146-147], the three of themnot lying in X ∪ Y are on a line (the white circles in Fig. 2B). Thus, by liaison (use formula 1.1)the set X ∪ Y is contained in a conic. (cid:3) Now we are ready to prove Theorem 3.1. We recall its statement.
Theorem 3.1
Let X = S (L) and Y = S (M) be two contact star configurations in P on thesame conic, where L = { (cid:96) , . . . , (cid:96) r } and M = { m , . . . , m s } are two sets of distinct lines. Then (a) if s = r − , then general form in ( I X ∪ Y ) r − is irreducible; (b) if s = r − , then X ∪ Y is a complete intersection of type ( r − , r − ) ; (c) if s = r , then the general form in ( I X ∪ Y ) r is irreducible; ATIONAL NORMAL CURVES AND HADAMARD PRODUCTS 7 (A)
Case r = s = . (B) The nine points intersectionof the two cubics.
Figure 2 (d) if s = r , let Z = Y ∩ m r , then ( I X ∪ Z ) r − = ( I X ∪ Y ) r − ; (e) if s = r , then X ∪ Y is a complete intersection of type ( r − , r ) .Proof. We proceed by induction on r . The cases r ≤ r >
3. Set X ( i ) = S (L ∖ { (cid:96) i }) , thus X ( i ) is a star configuration of points defined by r − X ( i ) ∪ Y , for i = , . . . , r −
1. By (e) and by induction, X ( i ) ∪ Y iscomplete intersection of type ( r − , r − ) , thus there exists a curve of degree r − X ( i ) ∪ Y , say C ( i ) r − .Note that C ( i ) r − does not have (cid:96) r as a component. In fact, if (cid:96) r is a component of C ( i ) r − ,then, by removing (cid:96) r , since it does not contain points of Y , we get a curve of degree r − Y . A contradiction, since I Y starts in degree r − . Since (cid:96) r is not a component for C ( i ) r − , then C ( i ) r − meets (cid:96) r in exactly r − ( X ∩ (cid:96) r ) ∖ ( (cid:96) i ∩ (cid:96) r ) .From here it easily follows that the linear system of curves(3.1) ⟨ (cid:96) i ∪ C ( i ) r − ∣ i = , . . . , r − ⟩ does not have any fixed component and it is not composite with a pencil. Thus byBertini’s Theorem the general curve in 3.1 is irreducible.(b) By (a), since the linear system (3.1) has dimension at least 2, there exist two irreduciblecurves of degree r − X ∪ Y . Since ∣ X ∪ Y ∣ = ( r ) + ( r − ) = ( r − ) , we are done.(c) From (a) the generic curve of degree r − X ( i ) ∪ Y is irreducible, say C ( i ) r − , foreach i = , . . . , r. Then, the linear system ⟨ (cid:96) i ∪ C ( i ) r − ∣ i = , . . . , r ⟩ does not have any fixedcomponent and it is not composite with a pencil. Again for Bertini’s Theorem we aredone.(d) Let P ir = (cid:96) i ∩ (cid:96) r and Y ( i ) = S (M ∖ { m i }) , i = , . . . , r −
1. From (b) the set X ∪ Y ( i ) is acomplete intersection of two curves of degree r − , hencedim k ( I X ∪ Y ( i ) ) r − = E. CARLINI, M. V. CATALISANO, G. FAVACCHIO, AND E. GUARDO and its h -vector is h X ∪ Y ( i ) = ( , , , . . . , r − , r − , r − , . . . , , ) . Moreover, Y ( i ) ∖ Z = S (M ∖ { m i , m r }) is a star configuration defined by r − h -vector is h Y ( i ) ∖ Z = ( , , , . . . , r − ) . Thus by liaison, see relation (1.1), we have h X ∪( Z ∖{ P ir }) = ( , , , . . . , r − , r − , r − ) , then dim k ( I X ∪( Z ∖{ P ir }) ) r − = . Since X ∪ ( Z ∖ { P ir }) ⊆ X ∪ Y ( i ) we have(3.2) ( I X ∪( Z ∖{ P ir }) ) r − = ( I X ∪ Y ( i ) ) r − . Note that in order to prove that dim ( I X ∪( Z ∖{ P ir }) ) r − =
2, we can use Lemma 2.2in [5] instead of liaison.Let F ∈ ( I X ∪ Z ) r − , then, by (3.2), F ∈ ( I X ∪ Y ( i ) ) r − for each i = , . . . , r −
1. So, F ∈ ( I X ∪ Y ) r − . It follows that ( I X ∪ Y ) r − = ( I X ∪ Z ) r − .(e) Let F ∈ ( I X ∪ Y ) r − as in the proof in item (d). By item (c) there exists an irreducibleform G ∈ ( I X ∪ Y ) r . Since ∣ F ⋅ G ∣ = r ( r − ) = ∣ X ∪ Y ∣ , then X ∪ Y is a complete intersection of the curves defined by F and G. (cid:3) A natural question related to Theorem 3.1 arises about the irreducibility of the curve of degree r − r = s . It cannot be guaranteed. Indeed, in the next example, we produce a setof 20 points (case r = s = Example 3.6.
Let
L = { (cid:96) , . . . , (cid:96) } be a set of five lines tangent to an irreducible conic γ (thegray parabola in Figure 3). The contact star configuration X = S (L) consists of ten points.We split the ten points in two sets of five points, each contained in an irreducible conic.Then, we consider the quartic union of these two conics (the one dashed and the other dottedin Figure 3).Now, we take a new line m tangent to γ , see Figure 4A. Through each of the four pointsintersection of m with the quartic, there is an extra tangent line to γ . Call these tangentliness m , m , m , m , see Figure 4B. The set of points Y = S ({ m , . . . , m , m }) is a contact starconfiguration and, from Theorem 3.1 (d), X ∪ Y is contained in the quartic. Of course thisquartic, by construction, is not irreducible.In order to show that the condition s = r or s = r − X ∪ Y to be a complete intersection, we prove the following lemma which holds with more generalassumptions. ATIONAL NORMAL CURVES AND HADAMARD PRODUCTS 9
Figure 3.
Example 3.6: The star configuration X and the not irreducible quar-tic through X . (A) The line m intersectingthe quartic in four points. (B) The contact star configu-ration Y . Figure 4.
Construction of the set of points X ∪ Y on a reduced quartic. Lemma 3.7.
Let X and Y be two disjoint star configurations defined by r and s lines, ( r ≥ s ),respectively. If X ∪ Y is a complete intersection, then either s = r or s = r − .Proof. Since h X = ( , , , . . . , r − ) and h Y = ( , , , . . . , s − ) , thus, by liaison, see formula (1.1),the h -vector of X ∪ Y must be h X ∪ Y = ( , , , . . . , r − , s − , . . . , , , ) . Since X ∪ Y is a complete intersection, then, to ensure the symmetry, we get s = r or s = r − (cid:3) Theorem 3.1 together with Lemma 3.7 give the following result.
Theorem 3.8.
Let X = S ( (cid:96) , . . . , (cid:96) r ) and Y = S ( m , . . . , m s ) be two contact star configurationson the same conic. Then X ∪ Y is a complete intersection if and only if either s = r or s = r − . The h -vector of union of contact star configurations in P In this section we work in P , so γ is always an irreducible conic, and we set S = C [ x, y, z ] = C [ P ] . The main result of this section, Theorem 4.3, shows that a union of two contact starconfigurations on a conic γ has the same h -vector of a scheme of two fat points in P . This pointof view will allows us to make further considerations on the h -vector of more than two contactstar configurations on a conic γ , see Theorem 4.5.The h -vector of two fat points is well known, several papers investigate it in a more generalsetting, see for instance [13, Theorem 1.5 and Example 1.6] and also [9, 11, 17, 18] just to citesome of them. Recall that if the multiplicities of the two points are m and n , with m ≥ n , thenthe h -vector is(4.1) ( , , . . . , m − , m, n, n − , . . . , , ) . In order to prove the claimed result, we need the following lemma.
Lemma 4.1.
Let Y = S ( m , . . . , m s ) be a star configurations of s lines, and let m s + , . . . , m s + t be t further lines, t ≥ . Set X = S ( m , . . . , m s , m s + , . . . , m s + t ) . Then, the h -vector of X ∖ Y is h X ∖ Y = ( , . . . , t − , t, t, . . . , t ·„„„„„„„„„„„„„‚„„„„„„„„„„„„„„¶ s ) . Proof.
Since the h -vector of X is ( , , . . . , s + t − ) and X ∖ Y is contained in a curve of degree t ,that is m t + ∪ ⋯ ∪ m t + s , then h X ∖ Y ≤ ( , . . . , t − , t, t, . . . , t ·„„„„„„„„„„„„„‚„„„„„„„„„„„„„„¶ s ) ≤ h X . Since, ∣ X ∖ Y ∣ = ( s + t ) − ( s ) = ( t ) + st = + ⋯ + t − + t + t + ⋯ + t ·„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„¶ s , we are done. (cid:3) The next example shows how we compute the h -vector of a union of two contact star config-urations in P . Example 4.2.
Let X = S ( (cid:96) , . . . , (cid:96) ) and Y = S ( m , m , m ) be two contact star configurationson a conic γ . Figure 5A gives a representation of this case.In order to compute the h -vector of X ∪ Y we consider three further lines m , m , m tangentto γ. Let Y ′ = S ( m , . . . , m ) ⊇ Y , see Figure 5B.From Theorem 3.1 (b), X ∪ Y ′ is a complete intersection of type ( , ) , hence the h -vector of X ∪ Y ′ is h X ∪ Y ′ = ( , , , , , , , , , , ) . By Lemma 3.7, the h -vector of Y ′ ∖ Y is h Y ′ ∖ Y =( , , , , ) . Thus, by formula (1.1), we get t h X ∪ Y ′ ( t ) h Y ′ ∖ Y ( − t ) h X ∪ Y ( t ) ATIONAL NORMAL CURVES AND HADAMARD PRODUCTS 11 (A)
The set X ∪ Y . (B) The set X ∪ Y ′ . Figure 5.
Example 4.2.which shows that X ∪ Y has the same h -vector of a scheme of two fat points of multiplicity 2and 6, see (4.1).Now we state the general theorem. Theorem 4.3.
Let X = S ( (cid:96) , . . . , (cid:96) r ) and Y = S ( m , . . . , m s ) be two contact star configurationson a conic. Let P, Q be two distinct points in P . Then h X ∪ Y = h ( r − ) P +( s − ) Q . Proof.
Assume s ≤ r . If either s = r or s = r −
1, then the statement follows from Theorem 3.1.In fact, in both the cases, X ∪ Y is a complete intersection and it has the required h -vector.So, assume s < r − t = r − s −
1. Consider t further lines, m s + , . . . , m s + t , tangent to γ ,and denote by Y ′ = S ( m , . . . , m s + t ) . From Theorem 3.1 (b), X ∪ Y ′ is a complete intersectionof type ( t + s, t + s ) and its h -vector is h X ∪ Y ′ = ( , , . . . , t + s − , t + s, t + s − , . . . , , ) . By Lemma 3.7, the h -vector of Y ′ ∖ Y is h Z = ( , , . . . , t − , t, t, . . . , t ·„„„„„„„„„„„„„‚„„„„„„„„„„„„„„¶ s ) . By formula (1.1), we get h X ∪ Y = ( , , . . . , t + s − , t + s, s − , s − , . . . , , ) == ( , , . . . , r − , r − , s − , s − , . . . , , ) which is the h -vector of ( r − ) P + ( s − ) Q . (cid:3) Theorem 3.1 allows us to compute the h -vector of a union of three contact star configurations,on a same conic in the special case described in Theorem 4.5. The proof of Theorem 4.5 requiresthe following well known result about the Hilbert function of a scheme of three fat points notlying on a line (see for instance [9, Theorem 3.1]). Proposition 4.4 (The Hilbert function of three fat points) . Let Z = m P + m P + m P be ascheme of three general fat points of multiplicity m ≥ m ≥ m ≥ , respectively. Then H Z ( d ) = { d + + H ′ Z ( d − ) if ≤ d ≤ m + m − ( Z ) if d ≥ m + m − where Z ′ = ( m − ) P + ( m − ) P + m P . Theorem 4.5.
Let X = S ( (cid:96) . . . , (cid:96) r ) , Y = S ( m , . . . , m s ) and W = S ( n , . . . , n t ) be contact starconfigurations on a conic, where t ≥ r ≥ s ≥ . Let Z = ( t − ) P + ( r − ) P + ( s − ) P be a schemeof three general fat points. If t ∈ { r + s − , r + s, r + s + } , then h X ∪ Y ∪ W = h Z (4.2) = ( , . . . , t − , t − , r + s − , r + s − , . . . , r − s + ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ s − , r − s, r − s − , r − s − , . . . ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ r − s ) . Proof.
We prove the theorem for t = r + s. The other two cases can be proved similarly.First we compute the h -vector of X ∪ Y ∪ W .Note that X ∪ Y is contained in the contact star configuration T = S ( m , . . . , m r , n , . . . , n s ) .Then T ∪ W , by Theorem 3.1 (d), is a complete intersection of type ( r + s − , r + s ) , and so h T ∪ W = ( , . . . , r + s − , r + s − , r + s − , r + s − , . . . , ) . Since the set T ∖ ( X ∪ Y ) is a complete intersection of type ( r, s ) , then its h -vector is ( , . . . , s − , s, . . . , s ·„„„„„„„„„„„‚„„„„„„„„„„„„¶ r − s + , s − , . . . , ) . Hence, by formula (1.1) we get(4.3) h X ∪ Y ∪ W = ( , . . . , r + s − , r + s − , r + s − , r + s − , . . . , r − s + ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ s − , r − s, r − s − , r − s − , . . . ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ r − s ) . Observe that, for s = h X ∪ Y ∪ W = ( , . . . , r, r + , r, r − , r − , . . . ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ r − ) , for r = s we have h X ∪ Y ∪ W = ( , . . . , r − , r − , r − , r − , . . . , ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ r − ) , and for r = s = h X ∪ Y ∪ W = ( , , , ) . We will prove the theorem by induction on 2 r + s −
3, that is, on the sum of the multiplicitiesof the three fat points. If 2 r + s − =
5, then Z = P + P + P , whose h -vector is ( , , , ) ,so the statement is proved for ( r, s ) = ( , ) . ATIONAL NORMAL CURVES AND HADAMARD PRODUCTS 13
Assume 2 r + s − ≥
7, and recall that Z = ( r + s − ) P + ( r − ) P + ( s − ) P ( r ≥ s ≥ H Z ( d ) = { d + + H Z ′ ( d − ) if 0 ≤ d ≤ ( r + s − ) + ( r − ) − ( Z ) if d ≥ ( r + s − ) + ( r − ) − Z ′ = ( r + s − ) P + ( r − ) P + ( s − ) P . Hence the h -vector of Z is(4.4) h Z ( d ) = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ + h Z ′ ( d − ) if 0 ≤ d ≤ ( r + s − ) + ( r − ) − ( Z ) − d − H Z ′ ( d − ) if d = ( r + s − ) + ( r − ) −
10 if d ≥ ( r + s − ) + ( r − ) . Now we will compute H Z ′ ( d − ) for d = ( r + s − ) + ( r − ) − = r + s − r > s we have r + s − > r − ≥ s −
1, and d − = r + s − = ( r + s − ) + ( r − ) −
1, which is thesum of the two highest multiplicities. Hence, by Proposition 4.4, we get H Z ′ ( r + s − ) = deg ( Z ′ ) .If r = s , we have Z ′ = ( r − ) P + ( r − ) P + ( r − ) P and we need to compute H Z ′ ( r − ) .Since the line P P is a fixed component for the curves of degree 3 r − Z ′ , we havedim ( I Z ′ ) r − = dim ( I Z ′′ ) r − , where Z ′′ = ( r − ) P + ( r − ) P + ( r − ) P . Since the scheme Z ′′ gives independent conditionsto the curve of degree 3 r − H Z ′′ ( r − ) = deg ( Z ′′ ) . It followsthat H Z ′ ( r − ) = ( r − ) − dim ( I Z ′ ) r − = ( r − ) − dim ( I Z ′′ ) r − = ( r − ) − ( r − ) + deg ( Z ′′ )= r − r + . Thus, for d = r + s − ( Z ) − d − H Z ′ ( d − ) = { deg ( Z ) − ( r + s − ) − deg ( Z ′ ) = r > s deg ( Z ) − ( r − ) − ( r − r + ) = r = s . From this equality and from (4.4) we get(4.5) h Z ( d ) = ⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩ + h Z ′ ( d − ) if 0 ≤ d ≤ r + s −
41 if d = r + s − r > s d = r − r = s d ≥ r + s − . Now, if r > s , from the inductive hypothesis, by substituting r with r − h -vector of Z ′ = ( r + s − ) P + ( r − ) P + ( s − ) P , that is, h Z ′ = ( , . . . , r + s − , r + s − , r + s − , r + s − , . . . , r − s + ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ s − , r − s − , r − s − , . . . )·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ r − s − . In case r = s , we have Z ′ = ( r − ) P + ( r − ) P + ( r − ) P , (note that 2 r − ≥ r − ≥ r − s ′ = r −
1. With this notation Z ′ = ( r + s ′ − ) P + ( r − ) P + ( s ′ − ) P . By applying theinductive hypothesis and then by substituting s ′ with r −
1, we get h Z ′ = ( , . . . , r + s ′ − , r + s ′ − , r + s ′ − , r + s ′ − , . . . , r − s ′ + ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ s ′ − , r − s ′ , r − s ′ − , r − s ′ − , . . . , ) = = ( , . . . , r − , r − , r − , r − , . . . , ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ r − , ) . By (4.3) and (4.5) the conclusion follows. (cid:3)
Remark 4.6.
Note that this theorem gives a non-algorithmic formula for the h -vector of threefat points of multiplicities m , m , m when m = m + m or m = m + m ± r = , s = , t = Example 4.7.
Let X = S ( (cid:96) , (cid:96) , (cid:96) ) , Y = S ( m , m , m ) and W = S ( n , . . . , n ) be contact starconfigurations on the same conic, see Figure 6A. Set T = S ( (cid:96) , (cid:96) , (cid:96) , m , m , m ) , see Figure 6B. (A) The union of three contactstar configurations on a conic. (B)
The nine points in T ∖( X ∪ Y ) on a grid. Figure 6.
Example 4.7.Then h W ∪ T = ( , , , , , , , , , ) ,h T ∖( X ∪ Y ) = ( , , , , ) , and, by liaison, h X ∪ Y ∪ W = ( , , , , , , ) , which is the h -vector of three fat points of multi-plicity 2 , , Conjecture 4.8.
The h -vector of a scheme of s ≤ general fat points of multiplicities m i , ( i = , . . . , s ) is equal to the h -vector of the union of s contact star configurations defined by m i + lines tangent to the same conic. The next example shows that the conjecture does not hold for s = Example 4.9.
The h -vector of five general fat points of multiplicity 2 in P is ( , , , , , ) but we checked with CoCoA [1] that the h -vector of five general contact star configurations ona conic defined by three lines is ( , , , , ) . ATIONAL NORMAL CURVES AND HADAMARD PRODUCTS 15 Applications of Theorem 3.1 to polygons
As an application of Theorem 3.1 we get a result that in a certain sense extend the Brianchon’sTheorem to an octagon circumscribed to a conic.
Proposition 5.1.
Let A , . . . , A be the vertices of an octagon and let (cid:96) ij be the line A i A j . Set P = (cid:96) ∩ (cid:96) and P = (cid:96) ∩ (cid:96) and let γ and γ be the conics through A , A , A , A , P and A , A , A , A , P , respectively. Let γ ∩ γ = { A , A , B , B } . If the octagon is circumscribed toa conic γ , then the points A , A , B , B are on a line. (See Figure 7) Figure 7.
The case described in Proposition 5.1
Proof.
Note that, by Theorem 3.1 (d), the point P = (cid:96) ∩ (cid:96) belongs to the conic γ . Infact, A , A , P and A , A , P are two contact star configurations on the same conic and thena complete intersection of type ( , ) . Analogously, the point P = (cid:96) ∩ (cid:96) belongs to theconic γ . Moreover, observe that the points A , A , A , P , P , P = (cid:96) ∩ (cid:96) , and the points A , A , A , P , P , P = (cid:96) ∩ (cid:96) are two contact star configurations each defined by four linestangent to the same conic, hence by Theorem 3.1 (d) these twelve points are a complete inter-section of type ( , ) , thus their h -vector is ( , , , , , ) . Now consider the two quartics γ ∪ (cid:96) ∪ (cid:96) and γ ∪ (cid:96) ∪ (cid:96) . This two quartics meet in acomplete intersection of sixteen points, that consists of the twelve points described above andthe points A , A , B , B . By relation (1.1), we get 0 1 2 3 4 5 6the h -vector of the sixteen points 1 2 3 4 3 2 1the h -vector of the twelve points 1 2 3 3 2 1the h -vector of { A , A , B , B } (cid:3) Corollary 5.2.
Let A , . . . , A be the vertices of an octagon and let (cid:96) ij be the line A i A j .Set P = (cid:96) ∩ (cid:96) , P = (cid:96) ∩ (cid:96) , P = (cid:96) ∩ (cid:96) , P = (cid:96) ∩ (cid:96) and let γ i be the conic through A i , A i + , A + i , A + i , P i , i = , . . . ( A = A ). If the octagon is circumscribed to a conic γ , thenthe eight points ( γ ∩ γ ) ∪ ( γ ∩ γ ) are on a conic. (See Figure 8). Figure 8.
The case described in Corollary 5.2
Proof.
The quartics γ ∪ γ and γ ∪ γ meet in sixteen points, in black and white in figure 5.2.Eight of them, the black dots, that is ( γ ∩ γ ) ∪ ( γ ∩ γ ) , by Proposition 5.1, lie on two lines.So, the h -vector of these eight points must be ( , , , , ) . By formula (1.1) the residual pointslie on a conic. (cid:3) Recall that, by Theorem 3.1(e), the six points union of two contact star configurations eachdefined by three line tangent to the same conic, are contained on a conic. An interesting caseoccurs considering three such contact star configurations, see Proposition 5.3, it can also betranslated in a property of a polygon of nine sides circumscribed to a conic.
Proposition 5.3.
Let X = S ( (cid:96) , (cid:96) , (cid:96) ) , X = S ( m , m , m ) , X = S ( n , n , n ) be contact starconfigurations on a conic. Let γ ij be the conic containing X i ∪ X j . Then, γ , γ , γ meet in apoint. See Figure 9A.Proof. Consider the cubic γ ∪ n and the quartic γ ∪ m ∪ m (respectively dotted and dashedin Figure 9B). The cubic and the quartic meet in twelve points (twice in m ∩ m ). Six of thesepoints, precisely X ∪ S ( n , m , m ) , lie on a conic, by Theorem 3.1(e). So, also the residual sixpoints lie on a conic (by formula 1.1) that is, by Theorem 3.1(d), the conic γ . (cid:3) Further directions
According to our computations, it should be possible to extend some of the results in thispaper to higher dimensional spaces. We state the following conjecture.
Conjecture 6.1.
Let X ∶= S ( (cid:96) , . . . , (cid:96) r ) and Y ∶= S ( m , . . . , m s ) be two contact star configu-rations in P n , where r ≥ s and all the hyperplanes are distinct. Then the h -vector of X ∪ Y is h X ∪ Y = ( , ( nn − ) , . . . , ( r − n − ) , ( s − n − ) , . . . , ( nn − ) , ) . In particular, if either s = r or s = r − then X ∪ Y is a Gorenstein set of points. ATIONAL NORMAL CURVES AND HADAMARD PRODUCTS 17 (A)
Three conics meet-ing in a point. (B)
The cubic and the quar-tic.
Figure 9.
Proposition 5.3.If Conjecture 6.1 is true then, it is possible to generalize Remark 3.3 in order to constructGorenstein sets of rational points in P n with a special h -vector. We show the procedure in thefollowing example. Example 6.2.
Let S = C [ x, y, z, t ] = C [ P ] and let (cid:96) ⊆ P be the line in P defined by the ideal ( z − x − y, t − x + y ) . Consider two sets of four points on (cid:96)X = {[ , , , ] , [ , , , ] , [ , , , ] , [ , , , ]} ,Y = {[ , − , , − ] , [ , − , − , − ] , [ , − , − , − ] , [ , − , − , − ]} . Then X ⋆ ∪ Y ⋆ = {[ , , , ] , [ , , , ] , [ , , , ] , [ , , , ]}∪{[ , − , , − ] , [ , − , , − ] , [ , − , , − ] , [ , − , − , − ]} . According to CoCoA, the set of eight points X ⋆ ∪ Y ⋆ ⊆ P is in fact Gorenstein and its h -vector is ( , , , ) . On the other hand, it is interesting to ask if contact star configurations need to be constructedon rational normal curves. Of course, one can extend the definition by taking, for instance, highcontact linear spaces to some other irreducible curve or surface and, with some assumptionsof generality, get again a star configuration. However, we don’t know if this construction on avariety of different kind will lead to configurations with special properties either from the pointof view of the h -vector or something else. So, we ask if the vice versa of Conjecture 6.1 is alsotrue. Question 6.3.
Let X ∶= S ( (cid:96) , . . . , (cid:96) r ) and Y ∶= S ( m , . . . , m s ) , where r ≥ s , be two star configu-rations in P n defined by distinct hyperplanes. Suppose that the h -vector of X ∪ Y is ( , ( nn − ) , . . . , ( rn − ) , ( sn − ) , . . . , ( nn − ) , ) . Then, are X and Y two contact star configurations on a same rational normal curve? A similar question can be asked in the case of Gorenstein set of points.
Question 6.4.
Let X ∶= S ( (cid:96) , . . . , (cid:96) r ) and Y ∶= S ( m , . . . , m s ) , where r ≥ s , be two star configu-rations in P n defined by distinct hyperplanes. Suppose that X ∪ Y is a Gorenstein set of pointsin P n . Then, are X and Y two contact star configurations on a same rational normal curve andeither s = r or s = r − ? In the next proposition we positively answer to Question 6.4 in P for the case r = s = . Proposition 6.5.
Let X and Y be two star configurations, both defined by distinct lines. Let X ∪ Y be a complete intersection of type ( , ) . Then X and Y are contact star configurationson the same conic γ .Proof. Let denote X = S ( (cid:96) , (cid:96) , (cid:96) ) and Y = S ( m , m , m ) . Set P ij ∶= (cid:96) i ∩ (cid:96) j and Q ij ∶= m i ∩ m j .Let denote by p ij and q ij the lines dual to P ij and Q ij and by L i and M j the points dual to thelines (cid:96) i and m j . By hypothesis there is a conic c passing thorough the six points in X ∪ Y . Then,the lines p ij and q ij are tangent to the conic c ∨ dual to c . Then { L , L , L } and { M , M , M } are c ∨ -contact star configurations. Hence, from Theorem 3.1 (e), there is a conic γ ∨ passingthrough { L , L , L , M , M , M } . This proves that X and Y are contact star configurations ona conic γ , that is, the conic dual to γ ∨ . (cid:3) References [1]
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Email address : [email protected] (M. V. Catalisano) Dipartimento di Ingegneria Meccanica, Energetica, Gestionale e dei Trasporti,Universit`a degli studi di Genova, Genoa, Italy
Email address : [email protected] (G. Favacchio) DISMA-Department of Mathematical Sciences, Politecnico di Torino, Turin, Italy
Email address : [email protected] (E. Guardo) Dipartimento di Matematica e Informatica, Universit`a degli studi di Catania, Cata-nia, Italy
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