Rational Normal Curves as Set-Theoretic Complete Intersections of Quadrics

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1 Rational Normal Curves as Set-Theoretic Complete Intersections of Quadrics Maria-Laura Torrente Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, I Genova, Italy Abstract In the first part of this paper we present a short survey on the problem of the representation of rational normal curves as set-theoretic complete intersections In the second part we use a method, introduced by Robbiano and Valla, to prove that the rational normal quartic is settheoretically complete intersection of quadrics: it is an original proof of a classical result of Perron, and Gallarati-Rollero Keywords: rational normal curves, set-theoretic complete intersection, quadratic polynomials, Gröbner bases 1 Introduction Mathematicians have always shown great interest for rational normal curves, a special class of curves obtained as the image of the projective line More precisely, the rational normal curve of degree n in P n, denoted by C n, is defined as the image of the Veronese embedding of P 1 in P n, which is given by the complete linear series of forms of degree n In particular, if we consider the vector space spanned by such linear forms and choose as its basis the set of monomials of degree n, we get the following parametric representation of C n : x 0 = a n, x 1 = a n 1 b,, x n 1 = ab n 1, x n = b n (1) Let s consider the special case n = 2 The parametric representation becomes x 0 = a 2, x 1 = ab, x 2 = b 2 It is easy to verify that the Cartesian representation of C 2 is given by the unique equation x 0 x 2 x 2 1 = 0, and that x 0 x 2 x 2 1 generates the ideal of the curve C 2, denoted by I(C 2 ) In this particular case everything is clear Now, we consider the next special case n = 3, that is, we consider the curve C 3 classically known as the twisted cubic Its parametric representation is: x 0 = a 3, x 1 = a 2 b, x 2 = ab 2, x 3 = b 3 The twisted cubic has degree 3 and codimension 2 in P 3 Further, it is easy to verify that C 3 is defined by more than 2 equations In fact, by contradiction,

2 2 Maria-Laura Torrente assume that it is defined by 2 equations, then, as a consequence of Bézout s theorem, C 3 would necessarily be the intersection of a plane and a surface of degree 3, contradicting the fact that C 3 is not a plane curve So, the curve C 3 is not a complete intersection, ie, the defining ideal I(C 3 ) of C 3 cannot be generated by as many equations as its codimension A minimal set of generators of I(C 3 ) can be computed in a standard way by using Gröbner bases theory (see for instance [8]): {x 0 x 2 x 2 1, x 0 x 3 x 1 x 2, x 1 x 3 x 2 2} We observe that such generators are the 2 2 minors of the matrix ( x0 x 1 x 2 x 1 x 2 x 3 ) Now, we consider the open set of the curve obtained by setting x 0 0; in this way, we obtain an affine chart of C 3, called Γ 3 Letting y i = xi x 0, for i = 1,, 3, the parametric representation of Γ 3 is y 1 = b, y 2 = b 2, y 3 = b 3 The ideal I(Γ 3 ) of the affine curve Γ 3 is generated by the polynomials f 1 = y 2 y1 2 and f 2 = y 3 y 1 y 2, that is, I(Γ 3 ) = (y 2 y1, 2 y 3 y 1 y 2 ), and so Γ 3 is a complete intersection But the curve Γ 3 can also be described by the two equations f 1 = 0 and f2 2 = 0; on the other hand, f 1 and f2 2 do not generate the ideal I(Γ 3 ), but obviously the curve Γ 3 is contained in their zero locus In fact we have that I(Γ 3 ) = (f 1, f2 2 ); more precisely this is equivalent to saying that f 1 and f2 2 define Γ 3 as a set-theoretic complete intersection Now, we observe that the ideal generated by f 1 and f2 2 is also generated by f 1 and g 2, where g 2 = f2 2 + y2f 2 1 = y3 2 2y 1 y 2 y 3 + y2 3 Using another well-known result in Gröbner bases theory (see for instance [9]), it is possible to prove that the zero locus of the homogenizations of f 1 and g 2 wrt x 0, which are F 1 = x 0 x 2 x 2 1 and G 2 = x 0 x 2 3 2x 1 x 2 x 3 + x 3 2 respectively, is exactly the curve C 3 We conclude that C 3 is a set-theoretic complete intersection, but it is not a complete intersection Is it possible to give a general statement for C n? This question has been answered in many different ways in the last seventy years In 1941 Perron observed that all rational normal curves C n are set-theoretical complete intersections; since then, many different contributions were devoted to improve the class of equations which define the curve C n set-theoretically But, why so great attention has been paid to this apparently minor issue? In our opinion, the reason is mainly due to the fact that the problem of characterizing which classes of projective varieties are set-theoretical complete intersections is not yet completely solved Any minor contribution to this topic would be of great importance; for this reason, in the last few years many mathematicians addressed this kind of problem, giving rise to a wide literature Among the others, we recall the papers of Barile [2 5], Badescu and Valla [1] (in particular, see Remark 43), Moh [10,11], Ohm [12], Robbiano [14, 15], Robbiano and Valla [16, 17], and Vogel [18, 19] This short survey paper, and in particular Section 2, is an excursus on the problem of the representation of the curve C n as a set-theoretic complete intersection In particular, throughout this paper we briefly recall the results contained in [7], [13], [17], [20] The contribution of [13] is fundamental: in this paper,

3 Rational Normal Curves as Set-Theoretic Complete Intersections of Quadrics 3 for the first time, it has been proved that every rational normal curve C n is the set-theoretical complete intersection of n 1 hypersurfaces of degree 2,, n respectively Further, in the special case when n is a power of 2, and only in this case, it is proved that C n is the set-theoretic complete intersection of n 1 quadrics Starting from this result and for every integer n, in [7] the authors find another representation of C n which lowers the degrees of the defining equations Let s be the largest power of 2 less or equal to n; then C n can be expressed as the set-theoretical complete intersection of s 1 quadrics and additional n s hypersurfaces of degree s + 1,, n respectively In our opinion, an alternative proof of the original results of [13], when n is a power of 2, can be obtained using the computational method presented in [17] (and reported in Section 2) which is a generalization of the example of the twisted cubic C 3 discussed in this introduction To this aim, in Section 3, we show that this latter method works for the case of the quartic curve C 4 ; its extension to the case n = 2 m is still an open issue 2 Classical results In this section we briefly recall some classical results on the problem of the representation of the rational normal curves as set-theoretic complete intersections Let C n be the rational normal curve of P n, and I n = I(C n ) be the defining ideal of C n in P = K[x 0, x 1,, x n ], the polynomial ring in the indeterminates x 0,, x n over the field K It is well-known that I n is generated by the 2 2 minors of the matrix ( ) x0 x A = 1 x n 1, x 1 x 2 x n ie, I n is generated by a system of ( n 2) quadratic forms A first answer to the representation problem is due to Perron (see [13]), who proved, in 1941, that the curve C n can be represented as the set-theoretic complete intersection of n 1 algebraic hypersurfaces of degree 2,, n and equations P 1 = 0,, P n 1 = 0 where x 0 x 1 x i x 1 x 2 x i+1 P i = (2) x i x i+1 x 2i for i = 1,, n 1, with x j = 0 for j > n The proof is direct and exploits the parametric representation of the rational normal curves: the core is to show that (x 0 : x 1 : : x n ) P n is a solution of the polynomial system P 1 = = P n = 0 if and only if it satisfies relation (1) When n is a power of 2, and only in this case, by making similar considerations and using an inductive approach, Perron shows that C n is a set-theoretic

4 4 Maria-Laura Torrente intersection of n 1 quadrics of equations Q n 1 = 0,, Q n n 1 = 0, recursively defined by: Q n 1 Q n i (x 0,, x n/2 ) i = 1,, n/2 1 i = Q n 1 i (x n/2+1,, x n ) i = n/2,, n 2 (3) n/2 k=0 ( 1)k( ) n/2 k xk x n k i = n 1 where the trivial case n = 2 is given by the quadric Q 2 1 = x 0 x 2 x 2 1 = 0 In this special case the curve C n can be represented by a system of polynomial equations whose degree, defined as the product of the degrees of the single equations, is equal to 2 n 1 In the late 1970s and early 1980s, independently of the work of Perron, two different proofs of the fact that the rational normal curves are set-theoretic complete intersections were presented in [17] and [20] In 1979, exploiting some properties of suitable homogeneous polynomials, Verdi (see [20]) proves that the prime ideal I n = I(C n ) satisfies I n (n 1)! J n, where J n is the ideal generated by the forms V 1,, V n 1 and x 0 x 1 x 2 x i x 1 x 2 x i+1 V i = x 2 0 (4) x i x i for i = 1,, n 1, and deg(v i ) = i + 1 Such a relation easily yields J n = I n In 1983, using computational techniques derived from Gröbner bases theory and an approach already introduced in [16], Robbiano and Valla (see [17]) provide a different proof of the fact that every curve C n is a set-theoretic complete intersection The method they use is more general and constructive: it addresses the problem of the representation of every projective variety V of codimension r in P n as a set-theoretic complete intersection, by performing the following steps: 1 Determine an affine variety W whose projective closure is V ; it is well-known that this implies that I(V ) = I(W ) hom, where I(W ) hom P denotes the homogeneous ideal generated by the homogenizations f hom of the polynomials f I(W ) wrt the homogenizing indeterminate x 0 (see for instance [9], Section 43) 2 Find a representation of W as a set-theoretic complete intersection, that is, determine polynomials g 1,, g r which satisfy (g 1,, g r ) = I(W ) 3 Lift the representation found in step 2 to the projective space to get a settheoretic complete intersection representation of V Note that the instructions of step 3 are very easy to be performed if the polynomials g 1,, g r are a Gröbner basis, wrt a degree compatible term ordering, of the ideal they generate J = (g 1,, g r ) In this case, using classical results

5 Rational Normal Curves as Set-Theoretic Complete Intersections of Quadrics 5 of Gröbner bases theory (see for instance [9], Proposition 4321), the homogenizations g1 hom,, gr hom are a Gröbner basis of the homogenized ideal J hom Further, recalling that over ideals the operations of homogenization and radical commute, we get the following chain of equalities: I(V ) = I(W ) hom = ( (g 1,, g r ) ) hom = (g 1,, g r ) hom = (g hom 1,, g hom r ) which implies that V is a set-theoretic complete intersection of the algebraic hypersurfaces of equations g1 hom = = gr hom = 0 Here we show how this computational approach has been used for the case of rational normal curves (see [17], Section 1) We consider the affine chart defined by x 0 0 and the monomial affine curve Γ n in A n whose parametric equations are x i = b i, for i = 1,, n It is clear that C n is the projective closure of Γ n, which implies that I n = I(C n ) = I(Γ n ) hom Further, it is easy to verify that Γ n is the complete intersection of n 1 hypersurfaces defined by polynomials f i = x i+1 x 1 x i, for i = 1,, n 1, from which it follows that I(Γ n ) = (f 1,, f n 1 ) Now, let 1 k i n 1; we have that x k 1 x k mod (f 1,, f i 1 ) and i ( ) i fi i = (x i+1 x 1 x i ) i = ( 1) k x i k i+1 k xk 1x k i k=0 i ( ) i x i i+1 + ( 1) k x i k i+1 k x kx k i mod (f 1,, f i 1 ) k=1 From the above relation and the definition of radical of an ideal (see for instance [8]) it follows that I(Γ n ) = (f 1,, f n 1 ) = (r 1,, r n 1 ), where r i are defined as follows: r i = x i i+1 + i ( ) i ( 1) k x i k i+1 k x kx k i (5) k=1 In K[x 1,, x n ], let σ be the total degree reverse lexicographical ordering induced by x 1 > σ > σ x n (denoted by DegRevLex in [8]); the leading term of each r i is LT σ (r i ) = x i+1 i Since the leading terms of r 1,, r n 1 form a regular sequence, it follows that r 1,, r n 1 are the Gröbner basis (wrt the term ordering σ) of the ideal they generate (r 1,, r n 1 ) (see again [8], Corollary 2510) We then conclude that the rational normal curve C n is the set-theoretic complete intersection of the hypersurfaces of equations r1 hom = 0,, rn 1 hom = 0 Note that the results of [13], [17] and [20] we have been discussing here yield different representations of C n In particular, we observe that, though deg(p i ) = deg(v i ) = deg(ri hom ) for each i, the homogeneous polynomials ri hom are in general much simpler (since they are formed by fewer terms) than the corresponding forms V i, which in turn are simpler than the forms P i (see also the following Example 1) Nevertheless, their common feature is the degree of

6 6 Maria-Laura Torrente the polynomial system they form, which is defined as the product of the degrees of all the polynomials and equals 2 3 n = n! in each of the three cases In 1988, partially using the results of [13], Gallarati and Rollero (see [7]) improve the representation of C n as a set-theoretic complete intersection from the point of view of the degree of the system Let s be the largest power of 2 less than or equal to n In [7] it is proved that there exists a system of polynomial equations defining C n n! 2s 1 and whose degree is s!, which is appreciably less than n! as soon as n 5 We give here a sketch of the proof Let Q s 1,, Q s s 1 be the the set of quadrics introduced in [13] and recursively defined in (3); further, let rs hom,, rn 1 hom be n s homogeneous forms of degree s+1,, n respectively (defined in (5)) A straightforward computation shows that (x 0 : : x n ) is a solution of the polynomial system { Q s i = 0 1 i s 1 ri hom = 0 s i n 1 (6) if and only if it satisfies relation (1) But this implies that the curve C n can be expressed as a set-theoretic complete intersection using the previous system of equations, whose degree is exactly n! 2s 1 s! We end this section with an example: the aim is to compute and compare the different representations of C n as a set-theoretic complete intersection discussed in this section Example 1 We consider the case of the rational normal quintic curve C 5 in P 5 In [13] the curve C 5 is expressed by the polynomials (see formula (2)): P 1 = x 2 1 x 0 x 2 P 2 = x 3 2 2x 1 x 2 x 3 + x 0 x x 2 1x 4 x 0 x 2 x 4 P 3 = x 4 3 3x 2 x 2 3x 4 + x 2 2x x 1 x 3 x 2 4 x 0 x x 2 2x 3 x 5 2x 1 x 2 3x 5 2x 1 x 2 x 4 x 5 + 2x 0 x 3 x 4 x 5 + x 2 1x 2 5 x 0 x 2 x 2 5 P 4 = x 5 4 4x 3 x 3 4x 5 + 3x 2 3x 4 x x 2 x 2 4x 2 5 2x 2 x 3 x 3 5 2x 1 x 4 x x 0 x 4 5 (7) In [20] the curve C 5 is expressed by the polynomials (see formula (4)): P 1 = x 2 1 x 0 x 2 V 2 = x 3 2 2x 1 x 2 x 3 + x 0 x 2 3 V 3 = x 4 3 3x 2 x 2 3x 4 + x 2 2x x 1 x 3 x 2 4 x 0 x 3 4 P 4 = x 5 4 4x 3 x 3 4x 5 + 3x 2 3x 4 x x 2 x 2 4x 2 5 2x 2 x 3 x 3 5 2x 1 x 4 x x 0 x 4 5 (8) In [17] the curve C 5 is expressed by the polynomials (see formula (5)): P 1 = x 2 1 x 0 x 2 V 2 = x 3 2 2x 1 x 2 x 3 + x 0 x 2 3 R 3 = x 4 3 3x 2 x 2 3x 4 + 3x 1 x 3 x 2 4 x 0 x 3 4 R 4 = x 5 4 4x 3 x 3 4x 5 + 6x 2 x 2 4x 2 5 4x 1 x 4 x x 0 x 4 5 (9)

7 Rational Normal Curves as Set-Theoretic Complete Intersections of Quadrics 7 In [7] the curve C 5 is expressed by the polynomials (see formula (6)): P 1 = x 2 1 x 0 x 2 G 2 = x 2 2 2x 1 x 3 + x 0 x 4 G 3 = x 2 3 x 2 x 4 R 4 = x 5 4 4x 3 x 3 4x 5 + 6x 2 x 2 4x 2 5 4x 1 x 4 x x 0 x 4 5 (10) As already observed, (7), (8) and (9) are made up of homogeneous polynomials of increasing degree 2, 3, 4, 5, so the corresponding polynomial systems have all degree 5! = 120 On the other hand, the elements of (10) are three quadratic polynomials and a quintic one, so the corresponding system has degree 40 Some polynomials (in particular the quadratic polynomial P 1 ) appear in various sets, but in general, moving from (7) to (10), the polynomial sets become more and more simple, where simplicity is measured according to the number of terms of the support of each polynomial In conclusion, the representation given by (10) seems to be the best one among the approaches we have considered here 3 The rational normal quartic in P 4 In this section we use the computational method introduced in [16, 17] (and recalled in Section 2) to provide a representation of the rational normal quartic as a set-theoretic complete intersection of quadrics Let C 4 in P 4 be the rational normal quartic, whose parametric representation is given by: x 0 = a 4, x 1 = a 3 b, x 2 = a 2 b 2, x 3 = ab 3, x 4 = b 4 Let I 4 = I(C 4 ) be the defining ideal of C 4 in P = K[x 0, x 1, x 2, x 3, x 4 ] We recall that I 4 is generated by the 2 2 minors of the matrix ( x 0 x 1 x 2 x 3 x 1 x 2 x 3 x 4 ), which are ϕ 12 = x 0 x 2 x 2 1 ϕ 13 = x 0 x 3 x 1 x 2 ϕ 14 = x 0 x 4 x 1 x 3 ϕ 23 = x 1 x 3 x 2 2 ϕ 24 = x 1 x 4 x 2 x 3 ϕ 34 = x 2 x 4 x 2 3 (11) We consider the affine chart defined by x 0 0; let y i = xi x 0, for i = 1,, 4, and let Γ 4 be the affine curve in A 4 defined parametrically by the equations: y 1 = b, y 2 = b 2, y 3 = b 3, y 4 = b 4 Finally, let I(Γ 4 ) be the prime ideal which defines Γ 4 in P = K[y 1, y 2, y 3, y 4 ] We observe that C 4 is the projective closure of Γ 4, which implies that I 4 = I(C 4 ) = I(Γ 4 ) hom The following proposition yields a representation of Γ 4 as a set-theoretic complete intersection Proposition 1 Let P = K[y 1, y 2, y 3, y 4 ], let Γ 4 in A 4 be the affine curve defined by the parametric equations y i = b i, for i = 1,, 4, and let I(Γ 4 ) be the prime ideal of P which defines Γ 4 Further, let f i = y i+1 y 1 y i, for i = 1,, 3, g = y 4 2y 1 y 3 + y 2 2, h = y 2 y 4 y 2 3, and let J = (f 1, f 2 2, g) The following equalities hold true:

8 8 Maria-Laura Torrente (a) I(Γ 4 ) = J; (b) J = (f 1, h, g) Proof To prove (a) we observe that I(Γ 4 ) is generated by (f 1, f 2, f 3 ), which is a regular sequence, implying that Γ 4 is a complete intersection Further, we point out that the following equality g = f 3 + y 2 f 1 y 1 f 2 holds true It follows that I(Γ 4 ) = (f 1, f 2, f 3 ) = (f 1, f 2, g), and so obviously I(Γ 4 ) = J To prove (b) it is sufficient to observe that f 2 2 = h + y 2 g y 2 2f 1 In the following theorem, we present an alternative constructive proof of the fact that C 4 is a set-theoretic complete intersection of quadrics Theorem 1 Notation as in Proposition 1 Let P = K[x 0, x 1, x 2, x 3, x 4 ], let C 4 be the rational normal quartic in P 4 and let I(Γ 4 ) be the ideal of P which defines C 4 Further, let Q = x 0 x 4 2x 1 x 3 + x 2 2 P Then: (a) I(Γ 4 ) hom = (ϕ 12, Q, ϕ 34 ), where ϕ 12, ϕ 34 are given in (11); (b) The curve C 4 is a set-theoretic complete intersection of the three quadrics of equations ϕ 12 = 0, Q = 0, ϕ 34 = 0 Proof By using Proposition 1 and the commutative property of the operations of homogenization and radical over ideals, we have I(Γ 4 ) hom = J hom = (f1, h, g) hom In P we fix the term ordering σ = DegRevLex (see [8]) and get LT σ (f 1 ) = y1, 2 LT σ (h) = y3, 2 LT σ (g) = y2, 2 which are pairwise coprime It follows that the polynomials f 1, h, g are the Gröbner basis (wrt the term ordering σ) of J Now, using Corollary 4320 of [9] and homogenizing wrt x 0 we get (f 1, h, g) hom = (f hom 1, h hom, g hom ) = (ϕ 12, Q, ϕ 34 ) which shows statement (a) Part (b) follows from statement (a) and the equality I(C 4 ) = I(Γ 4 ) hom Acknowledgements I would like to thank Prof D Gallarati for bringing my attention to his paper [7], a joint work with Prof A Rollero, allowing me to write these notes I deeply thank Prof L Robbiano and Prof M C Beltrametti for numerous helpful discussions on the topic References 1 Badescu, L, Valla, G: Grothendieck-Lefschetz theory, set-theoretic complete intersections and rational normal scrolls J Algebra 324, (2010) 2 Barile, M: Certain minimal varieties are set-theoretic complete intersections Comm Algebra 35(7), (2007) 3 Barile, M: On binomial set-theoretic complete intersections in characteristic p Rev Mat Complut 21(1), (2008)

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