A generalized stick figure is a union of linear subvarieties of

156 C. Bocci - G. Dalzotto complete intersection.

2. Gorenstein points in

P 3 from the h-vector In this section we will see how Nagel and Migliore find a reduced arithmetically Gorenstein zeroscheme in P 3 i.e. a reduced Gorenstein quotient of k[x , x 1 , x 2 , x 3 ] of Krull dimension 1 with given h-vector. We start with some basic definitions that we find in [6] and in [9]. D EFINITION 1. Let H = h , h 1 , . . . , h i , . . . be a finite sequence of non-negative inte- gers. Then H is called an O-sequence if h = 1 and h i+1 ≤ h i i for all i . By the Macaulay theorem we know that the O-sequences are the Hilbert functions of stan- dard graded k-algebras. D EFINITION 2. Let h = 1, h 1 , . . . , h s−1 , 1 be a sequence of non-negative integers. Then h is an SI-sequence if: • h i = h s−i for all i = 0, . . . , s, • h , h 1 − h , . . . , h t − h t −1 , 0, . . . is an O-sequence, where t is the greatest integer ≤ s 2 . Stanley [9] characterized the h-vectors of all graded Artinian Gorenstein quotients of k[x , x 1 , x 2 ], showing that these are SI-sequence and any SI-sequence, with h 1 = 3, is the h- vector of some Artinian Gorenstein quotient of k[x , x 1 , x 2 ]. Now we can see how Nagel and Migliore [6] find a reduced arithmetically Gorenstein ze- roscheme in P 3 with given h−vector. This set of points will result from the intersection of two arithmetically Cohen-Macaulay curves in P 3 , linked by a Complete Intersection curve which is a stick figure. D EFINITION

3. A generalized stick figure is a union of linear subvarieties of

P n , of the same dimension d, such that the intersection of any three components has dimension at most d − 2 the empty set has dimension -1. In particular, sets of reduced points are stick figure, and a stick figure of dimension d = 1 is nothing more than a reduced union of lines having only nodes as singularities. So, let h = h , h 1 , . . . , h s = 1, 3, h 2 , . . . , h t −1 , h t , h t , . . . , h t , h t −1 , . . . , h 2 , 3, 1 be a SI-sequence, and consider the first difference 1 h = 1, 2, h 2 − h 1 , . . . , h t − h t −1 , 0, 0, . . . , 0, h t −1 − h t , . . . , − 2, −1 Define two sequences a = a , . . . , a t and g = g , . . . , g s+1 in the following way: a i = h i − h i−1 f or 0 ≤ i ≤ t Gorenstein points in P 3 157 and g i =      i + 1 for 0 ≤ i ≤ t t + 1 for t ≤ i ≤ s − t + 1 s − i + 2 for s − t + 1 ≤ i ≤ s + 1 We observe that a 1 = g 1 = 2, a is a O-sequence since h is a SI-sequence and g is the h- vector of a codimension two Complete Intersection. So, we would like to find two curves C and X in P 3 with h-vector respectively a and g. In particular it is easy to see that g is the h-vector of a Complete Intersection, X , of two surfaces in P 3 of degree t + 1 and s − t + 2. We can get X as a stick figure by taking as the equation of those surfaces two forms which are the product, respectively, of A , . . . , A t and B , . . . , B s−t +1 , all generic linear forms. Nagel and Migliore [6] proved that the stick figure embedded in X , determined by the union of a i consecutive lines in A i = 0 always the first in B = 0, is an aCM scheme C with h-vector a. In this way, if we consider C ′ , the residual of C in X , the intersection of C and C ′ is an aG scheme Y of codimension 3. This is also a reduced set of points because X , C and C ′ are stick figures and it has the desired h-vector by the following theorem: T HEOREM 1. Let C, C ′ , X , Y be as above. Let g = 1, c, g 2 , . . . , g s , g s+1 be the h-vector of X , let a = 1, a 1 , . . . , a t and b = 1, b 1 , . . . , b l be the h-vectors of C and C ′ , then b i = g s+1−i − a s+1−i for i ≥ 0. Moreover the sequence d i = a i + b i − g i is the first difference of the h-vector of Y . So we have to show that d i = h i − h i−1 : • For 0 ≤ i ≤ t we have d i = a i = h i − h i−1 • For t + 1 ≤ i ≤ s − t we have d i = b i − g i = 0 • For s − t + 1 ≤ i ≤ s + 1 we have d i = b i − g i = −a s+1−i = −h s+1−i − h s−i = h i − h i−1 R EMARK 1. Theorem 1 says, for example, that there exists no cubic through the 8 points of a Complete Intersection of two cubics, but not through the nine. In fact, if we consider a reduced Complete Intersection zeroscheme X in P 2 given by two forms of degree a and b, the h−vector of X \ { P} is 1, 2, 3, . . . , a − 1, a, a, . . . , a, a, a − 1, . . . , 3, 2, whatever point P we cut off. E XAMPLE 1. Let h = 1, 3, 4, 3, 1 be a SI-sequence. Consider the first difference of h, i.e. 1h = 1, 2, 1, −1, −2, −1. So, g = 1, 2, 3, 3, 2, 1 is the h-vector of X , stick figure which is the Complete Intersection of F 1 = Q 2 i=0 A i and F 2 = Q 3 i=0 B i , where A i and B i are general linear forms. Now, we call P i, j the intersection between A i = 0 and B j = 0. Then C = P 0,0 ∪ P 1,0 ∪ P 1,1 ∪ P 2,0 is the scheme which has h-vector a = 1, 2, 1. 158 C. Bocci - G. Dalzotto ✈ ❢ ❢ ❢ ✈ ✈ ❢ ❢ ✈ ❢ ❢ ❢ ✈ ❢ C = C ′ = P 0,0 P 0,1 P 0,2 P 0,3 P 1,0 P 1,1 P 1,2 P 1,3 P 2,0 P 2,1 P 2,2 P 2,3 Figure 1 A 2 A 1 A B 3 B 2 B 1 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . So, it is clear that the residual C ′ of C in X is the union of the lines of X which aren’t components in C. Then the reduced set of points Y with h-vector 1, 3, 4, 3, 1 consists of 12 points which exactly are: • 3 points on P 0,0 , intersection between P 0,0 and P 0,1 , P 0,2 and P 0,3 • 2 points on P 1,0 , intersection between P 1,0 and P 1,2 , P 1,3 • 4 points on P 1,1 , intersection between P 1,1 and P 1,2 , P 1,3 , P 0,1 and P 2,1 • 3 points on P 2,0 , intersection between P 2,0 and P 2,1 , P 2,2 and P 2,3 E XAMPLE 2. Let h = 1, 3, 5, 3, 1. With the previous notations, we have that the first difference of h is 1h = 1, 2, 2, −2, −2, −1, so g = 1, 2, 3, 3, 2, 1. Hence, we can take a stick figure X which is a Complete Intersection between a cubic and a quartic. Therefore, as above, we get a subscheme of X with h-vector 1, 2, 2. ✈ ✈ ❢ ❢ ✈ ✈ ❢ ❢ ✈ ❢ ❢ ❢ ✈ ❢ C = C ′ = A 2 A 1 A B 3 B 2 B 1 B Figure 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . In this way, the intersection between C and the residual C ′ gives the reduced set of 13 points with the expected h-vector. Gorenstein points in P 3 159

3. Gorenstein Sets of points not complete intersection