Abstract
The type II 1 unprojection is, by definition, the generic complete intersection type II unprojection, in the sense of [P] Section 3.1, for the parameter value k = 1, and depends on a parameter n ≥ 2. Our main results are the explicit calculation of the linear relations of the type II 1 unprojection for any value n ≥ 2 (Theorem 3.16) and the explicit calculation of the quadratic equation for the case n = 3 (Theorem 4.1). In addition, Section 5 contains applications to algebraic geometry, while Section 6 contains the Macaulay 2 code for the type II 1 unprojection for the parameter value n = 3.
Introduction
The first appearance of the type II unprojection was in the study of elliptic involutions between Fano 3-fold hypersurfaces in [CPR] , while the theoretical foundations were developed in [P] using valuations. Valuations were useful to prove the existence of certain relations, but didn't provide any explicit formulas for them. The present paper is an effort towards the explicit calculations for the type II unprojection.
Introduction
We define the type II k unprojection to be the generic complete intersection type II unprojection, in the sense of [P] Section 3.1, for the parameter value k ≥ 1. According to [P] , it depends on a parameter n ≥ 2, increases the codimension from nk − 1 to nk − 1 + (k + 1) and preserves Gorensteiness. To our knowledge, the only previously done explicit calculation for the type II unprojection was for the type II 1 case for n = 2 ( [CPR] , [R] ), which is reproduced for completeness in Subsection 4.4 below.
Introduction
Section 3 contains the calculation, using homological and multilinear algebra, of the linear equations of the type II 1 unprojection for any value n ≥ 2. The main result is Theorem 3.16, which provides explicit formulas for the linear relations.
Introduction
Section 4 contains the calculation of the quadratic equation of the type II 1 unprojection for the case n = 3. The main result is Theorem 4.1 which provides an explicit symmetric -in the sense of SL 3 invariance, see Subsection 4.2 -formula for the quadratic relation for this case. The proof of Theorem 4.1 is based on the explicit equality (4.6) which was verified using the computer algebra program . In Subsection 4.3 we briefly sketch how we arrived to the formulas contained in Theorem 4.1 by computer assisted calculations (using the computer algebra program Maple).
Introduction
As an application of the above results, we sketch in Section 5 the construction of two codimension 4 Fano 3-folds. The quasismoothness checking, using the explicit equations obtained in Sections 3 and 4, was done by the computer algebra program Singular [GPS01] .
Introduction
The calculation of the quadratic equation for the type II 1 unprojection for n ≥ 4 remains open. We believe that the explicit formulas of the linear relations obtained in the present work together with representation and invariant theoretic techniques could, perhaps, lead to their calculation, and also to a better understanding of the complicated looking formula (4.2). To our knowledge, the problem of calculating the type II k unprojection for k ≥ 2 is also open.
Introduction
ACKNOWLEDGEMENTS: I wish to thank Janko Boehm, Stephen Donkin, Sébastien Jansou, Nondas Kechagias, Frank-Olaf Schreyer and Bart Van Steirteghem for useful discussions. Parts of this work were financially supported by the Deutschen Forschungsgemeinschaft Schr 307/4-2 and by the Portuguese Fundação para a Ciência e a Tecnologia through Grant SFRH/BPD/22846/2005.
Notation
As already mentioned in Section 1 by type II 1 unprojection we mean the generic complete intersection type II unprojection in the sense of [P] Section 3.1 with fixed parameter value k = 1. According to [P] , the type II 1 unprojection depends on an integral parameter n ≥ 2, the initial data for the unprojection is specified by a triple where 1 ≤ p ≤ n − 1, 1 ≤ i < j ≤ n and 1 ≤ l ≤ m ≤ n. The ring O amb corresponds to the ring denoted by O g amb in [P] Section 3.1, the variable x i corresponds to the variable a 1i in [P] Section 2, and the variable y i corresponds to the variable a 2i in [P] Section 2. (In Section 4 O amb will change slightly to allow 2 to be invertible.)
Notation
We denote by M the 2 × 2n matrix
Notation
which corresponds to the matrix with the same name in [P] Equation (2.1).
Notation
The ideal I D ⊂ O amb in (2.1) is the ideal generated by the 2 × 2 minors of M and corresponds to the ideal I g D in [P] Section 3.1. For 1 ≤ p ≤ n − 1, we denote by A p the n × n skew-symmetric matrix with (ij)-entry (for i < j) equal to A p ij (and zero diagonal entries), and by B p the n × n symmetric matrix with (ij)-entry (for i ≤ j) equal to B p ij . For 1 ≤ p ≤ n − 1 we set
Notation
where,
Notation
for 1 ≤ i < j ≤ n and 1 ≤ m ≤ n. In matrix notation
Notation
The ideal I X in (2.1) is the ideal
Notation
and corresponds to the ideal I g X in [P] Section 3.1. In addition, we set x = (x 1 , . . . , x n ), y = (y 1 , . . . , y n ).
Notation
(2.5) In other words, we have the equality
Notation
. Calculating I Y means providing explicit formulas for l 1 , l 2 , . . . , l 2n and q.
Notation
For simplicity, we call the elements of I X the original relations, we call l 1 , . . . , l 2n the linear relations and, finally, we call q the quadratic relation.
Notation
For the parameter value n = 2 the ideal I Y has been explicitly calculated by Reid in [R] Section 9.5. For completeness, we reproduce his results in Subsection 4.4.
Notation
In Theorem 3.16 we calculate, for any n ≥ 2, the linear relations l 1 , . . . , l 2n . In addition, in Theorem 4.1 we calculate the quadratic polynomial q for the parameter value n = 3.
Notation
The calculation of the quadratic polynomial q for n ≥ 4 remains open.
Notation
3 Linear relations of Type II 1 unprojection
Notation
In this section we use the notations of Section 2.
Generalities
Assume R is a ring, L is an R-module, and h :
Generalities
associated to the homomorphism h.
Generalities
Proposition 3.1 Assume h 1 , h 2 : L → R are two homomorphisms of Rmodules. For p ≥ 2 we have the equality
Generalities
Proof Indeed,
The second complex
Assume L is a free O amb -module of rank n, and let e 1 , . . . , e n be a fixed basis of L. We define four
The second complex
In addition, we define, for 1 ≤ p ≤ n, homomorphisms
The second complex
Proposition 3.2 For p ≥ 2 we have
The second complex
Proof Using the equalities h 4 = h 1 and h 3 = zh 2 the result follows from Proposition 3.1. QED
The second complex
The proof of the following proposition will be given in the Subsection 3.3.
The second complex
is exact.
Proof of Proposition 3.3
The proof of Proposition 3.3 will be based on the Buchsbaum-Eisenbud acyclicity criterion as stated in [BH] Theorem 1.4.13. We first need the following combinatorial lemma.
Proof of Proposition 3.3
Lemma 3.4 Let p be an integer with 1 ≤ p ≤ n. Then
Proof of Proposition 3.3
Proof We have
Proof of Proposition 3.3
where we used the Pascal's rule
Proof of Proposition 3.3
which is valid whenever 1 ≤ j ≤ n − 1. QED For the following we fix an integer p with 1 ≤ p ≤ n. We set
Proof of Proposition 3.3
where for a finitely generated free O amb -module N we denote by rank N the minimal number of generators of N as O amb -module. Using Lemma 3.4 we get
Proof of Proposition 3.3
Denote by I rp (φ p ) the r p -th Fitting ideal of the map φ p , that is the ideal of O amb generated by the r p × r p minors of any matrix representation of φ p . (For more information about Fitting ideals see, for example, [BH] Section 1.4.)
Proof of Proposition 3.3
We will need the following general property of Koszul complexes.
Proof of Proposition 3.3
Lemma 3.5 Fix t ∈ {1, 2} and s, p with 1 ≤ s, p ≤ n. Denote by M p t the matrix representation of the map
Proof of Proposition 3.3
For a suitable ordering of the two bases there exists an (r p /2) × (r p /2) submatrix of M p t which is diagonal with diagonal entries equal to a t or −a t , where a 1 = y s and a 2 = x s .
Proof of Proposition 3.3
Proof Consider the subset of the above mentioned basis of ∧ p L consisting of the elements where e s appears on the wedge. This subset has n−1 p−1 elements, hence (r p /2) elements using (3.3). The result follows by restricting the homomorphism d p h t to the O amb -submodule of ∧ p L generated by this subset and noticing that the corresponding matrix has a diagonal submatrix of the desired form. QED Proposition 3.6 Fix s, p with with 1 ≤ s, p ≤ n. There exists a polynomial g s ∈ I rp (φ p ) of the form
Proof of Proposition 3.3
where g 1 s involves only the variables y s , z, x s and has degree in y s strictly less than r p .
Proof of Proposition 3.3
Proof It follows immediately from the definition of φ p and Lemma 3.5. QED Proposition 3.7 Let F be either the field Q of rational numbers or a finite field Z/(q), where q is a prime. The ideal I of O amb ⊗ Z F generated by the image of I rp (φ p ) under the natural map O amb → O amb ⊗ Z F has codimension greater or equal than n.
Proof of Proposition 3.3
Proof Using Proposition 3.6 it is clear that there exists a monomial ordering for the variables y j , z, x j (with indices 1 ≤ j ≤ n) such that the leading term of the polynomial g s appearing in (3.4) is equal to y Proposition 3.8 The ideal I rp (φ p ) has grade greater or equal than n.
Proof of Proposition 3.3
Proof Combining Proposition 3.7 and [L] Theorem 3.12 we get that I rp (φ p ) has codimension in O amb greater or equal than n. Being a polynomial ring over the integers, the ring O amb is Cohen-Macaulay. Using [BH] Corollary 2.14 the result follows. QED We now finish the proof of Proposition 3.3. Combining Lemma 3.4 and Proposition 3.8, Proposition 3.3 follows from the Buchsbaum-Eisenbud acyclicity criterion as stated in [BH] Theorem 1.4.13.
The first complex
Fix n ≥ 2. Let N be a free O amb -module of rank n − 1, with basis e 1 , . . . , e n−1 .
The first complex
We define an O amb -homomorphism ψ : N → O amb , with ψ( e p ) = f p , for 1 ≤ p ≤ n − 1, where f p was defined in (2.4). As in Subsection 3.1, we have the Koszul complex
The first complex
Proposition 3.9 The sequence f 1 , . . . , f n−1 is an O amb -regular sequence. As a consequence, the complex N * is exact.
The first complex
Proof For an element g of O amb we denote by η(g) the result of substituting to g zero for z, zero for A p ij with 1 ≤ p ≤ n − 1 and 1 ≤ i < j ≤ n, and zero for B 
The first complex
Therefore, the ideal (η(f 1 ), . . . , η(f n−1 )) ⊂ O amb has codimension in O amb equal to n − 1. As a consequence, the ideal (f 1 , . . . , f n−1 ) of O amb has also codimension n−1. Since O amb is Cohen-Macaulay the sequence f 1 , . . . , f n−1 is an O amb -regular sequence. The exactness of the Koszul complex N * follows from [BH] Corollary 1.6.14. QED
The first commutative square
We define two
The first commutative square
That is, they will satisfy the two conditions
The first commutative square
We set, for 1 ≤ j ≤ m,
The first commutative square
c ij e i and
The first commutative square
In matrix notation, u 1 and u 2 are given, with respect to the above defined bases ( e j ) of N and (e i ) of L, by the following two n × (n − 1) matrices:
The first commutative square
That is, for 1 ≤ j ≤ n − 1, u 1 is the matrix with j-th column equal to −A j x t + B j y t , while u 2 is the matrix with j-th column equal to −B j x t .
The first commutative square
Proposition 3.10 The maps u 1 and u 2 satisfy the equations (3.6) and (3.7).
The first commutative square
Proof An easy calculation using (3.8). QED
The basic formula
The key result of the present subsection is Proposition 3.12, essentially a formal consequence of (3.7), which relates the differentials and the alternating products of the homomorphisms h i and u j . We need the following definitions.
The basic formula
for all c 1 , . . . , c p ∈ N . We denote the map ∧ p a 1 also by alt(a p 1 ). We will now define for p, q ≥ 1 an O amb -homomorphism
The basic formula
We set alt(a
The basic formula
where the sum is over all subsets I ⊆ {1, . . . , p + q}, with |I| = p and, by definition, w I = w 1 ∧ · · · ∧ w p+q with w i = a 1 if i ∈ I and w i = a 2 if i / ∈ I. More precisely, even though the tensor product map
The basic formula
does not induce a map ∧ p+q N → ∧ p+q L, taking the sum of those maps over the family of subsets I as above induces a well-defined map ∧ p+q N → ∧ p+q L, and this is the map denoted by alt(a
The basic formula
Remark 3.11 For p = q = 1 we have
The basic formula
for all c 1 , c 2 ∈ N . We will also denote alt(a 1 , a 2 ) by
The basic formula
For the following proposition, which will be proved in Subsection 3.7, recall that u 1 , u 2 : N → L were defined in Subsection 3.5, and h 1 , h 2 : L → O amb were defined in Subsection 3.2. We also use the notational conventions alt(u
The basic formula
2 ) = 0 for p = 1 and q ≥ 1, and alt(u 2 ) = 0 for q = 1 and p ≥ 1.
The basic formula
where the equality is, of course, as maps
Proof of Proposition 3.12
We will use the following two lemmas.
Proof of Proposition 3.12
Lemma 3.13 Assume p, q ≥ 1 and u 1 , u 2 : N → L are two O amb -module homomorphisms. We have
Proof of Proposition 3.12
in the sense that
Proof of Proposition 3.12
for all c ∈ N and c 1 ∈ ∧ p+q−1 N .
Proof of Proposition 3.12
Proof Immediate from the definitions. QED
Proof of Proposition 3.12
for all c ∈ N and c 1 ∈ ∧ p+q N .
Proof of Proposition 3.12
Proof Since
Proof of Proposition 3.12
we have, using Lemma 3.13, that
Proof of Proposition 3.12
and the result follows. QED
Proof of Proposition 3.12
We will now prove part a) of Proposition 3.12. Assume p ≥ 2. For c 1 , . . . , c p−1 ∈ N we have
Proof of Proposition 3.12
We will now prove part b) of Proposition 3.12. Assume that p, q ≥ 1, that N, L are O amb -modules and that u 1 , u 2 : N → L and h 1 , h 2 : L → O amb are four O amb -module homomorphisms with the property
Proof of Proposition 3.12
cf. (3.7). We will show by induction on p + q that (3.9) holds. If p = q = 1 the result is clear, since it is exactly our hypothesis (3.11). Assume now that p + q ≥ 3 and that (3.9) holds for the (p − 1, q) and (p, q − 1) cases.
Proof of Proposition 3.12
Using Lemma 3.13 we have
Proof of Proposition 3.12
which, using (3.10), is equal to
Proof of Proposition 3.12
which, using (3.11), is equal to
Proof of Proposition 3.12
which, using the inductive hypothesis, is equal to
Proof of Proposition 3.12
which, using Lemma 3.14, is equal to
Proof of Proposition 3.12
which finishes the proof of Proposition 3.12.
The connecting homomorphisms
For an integer p, with 1 ≤ p ≤ n − 1, we set
The connecting homomorphisms
with the summation as long as all exponents are nonnegative. More precisely, using the notational convention z 0 = 1, and denoting, for j = 1, 2, by T j p the j-th row of T p , we have
The connecting homomorphisms
).
The connecting homomorphisms
We consider
The connecting homomorphisms
For the following proposition recall that the map φ p was defined in Subsection 3.2 while the map ψ was defined in Subsection 3.4.
Proposition 3.15
Proof Fix p ≥ 2. For i ∈ Z we set
Proposition 3.15
if 0 ≤ i ≤ p and C i = 0 otherwise, and we set
Proposition 3.15
For the rest of the proof all sums are for i ∈ Z. We have
Proposition 3.15
By Proposition 3.12, for any i ∈ Z we have
Proposition 3.15
Using (3.6), we need to show that
Proposition 3.15
and that
Proposition 3.15
Using (3.12) we get
QED

Linear equations of type II 1 unprojection
Assume n ≥ 2 is given. We use the notations of the previous subsections. In Subsection 3.8 we defined two O amb -module homomorphisms T 1 n−1 , T 2 n−1 :
Linear equations of type II 1 unprojection
There exist, for j = 1, 2 and 1 ≤ i ≤ n, unique element σ
Linear equations of type II 1 unprojection
For 1 ≤ i ≤ n we define the polynomials l i , l i+n ∈ O amb [s 0 , s 1 ] with
Linear equations of type II 1 unprojection
Using Proposition 3.15 and arguing as in [R] Section 9.5 we get the following Theorem. 4 Quadratic relation of Type II 1 for n = 3
Linear equations of type II 1 unprojection
Assume now that we are in the type II 1 unprojection with n = 3. We follow the notations of Section 2 with only one change: Since our symmetric formula (4.2) for the quadratic relation will need 1/2 as coefficient, our ambient ring O amb will now be
Linear equations of type II 1 unprojection
with indices 1 ≤ p ≤ 2 and 1 ≤ i < j ≤ 3, 1 ≤ l ≤ m ≤ 3.
Linear equations of type II 1 unprojection
Recall that in Section 2 we defined two symmetric 3 × 3 matrices B 1 and B 2 by
Linear equations of type II 1 unprojection
for j = 1, 2, two skew-symmetric 3 × 3 matrices A 1 and A 2 by
Linear equations of type II 1 unprojection
for j = 1, 2, and two 1 × 3 matrices x, y with x = (x 1 , x 2 , x 3 ), y = (y 1 , y 2 , y 3 ).
Linear equations of type II 1 unprojection
We now define three induced 3 × 3 symmetric matrices ad B 1 , ad B 2 and ad B 1,2 .
Linear equations of type II 1 unprojection
For a square n × n matrix M with entries in a commutative ring R with unit, we denote by ad M the n × n matrix with kl-th entry, for 1 ≤ k, l ≤ n, equal to (−1) k+l times the determinant of the submatrix of M obtained by deleting the l-th row and the k-th column. It is well-known that we have the identities M (ad M ) = (ad M )M = (det M )I n , where I n is the identity n × n matrix over R.
Linear equations of type II 1 unprojection
In our situation, since B 1 and B 2 are symmetric 3 × 3 matrices we have that ad B 1 and ad B 2 are symmetric 3 × 3 matrices, such that for j = 1, 2 ad Notice that for 1 ≤ k ≤ 3, 1 ≤ l ≤ 3, we have
Linear equations of type II 1 unprojection
mj ) where {i, k, m} = {j, l, n} = {1, 2, 3} and i < m, j < n, and similarly for ad B 2 .
Linear equations of type II 1 unprojection
We define ad B 1,2 to be the 3 × 3 symmetric matrix, such that, for 1 and the rest of the entries of ad B 1,2 are determined by symmetry.
Linear equations of type II 1 unprojection
We now define two polynomials a, b ∈ O amb . We set 
Linear equations of type II 1 unprojection
The two polynomials f 1 , f 2 corresponding to the ones in (2.4), are
Linear equations of type II 1 unprojection
3)
Linear equations of type II 1 unprojection
for j = 1, 2. We denote by u 1 the 3 × 2 matrix
Linear equations of type II 1 unprojection
and by u 2 the 3 × 2 matrix
Linear equations of type II 1 unprojection
These matrices correspond to the maps with the same name defined in (3.8).
Linear equations of type II 1 unprojection
For j = 1, 2, we set φ 1 (u j ) to be the 3 × 1 matrix
Linear equations of type II 1 unprojection
and we also define the 3 × 1 matrix The matrix φ 1 (u j ) corresponds to the map ∧ 2 u j and the matrix φ 2 (u 1 , u 2 ) corresponds to the map alt(u 1 , u 2 ) which were defined in Subsection 3.6. We denote, for 1 ≤ i ≤ 3,
Linear equations of type II 1 unprojection
where we use the usual notation M ij for the ij-th entry of a matrix M . Clearly, for 1 ≤ i ≤ 6, l i ∈ O amb [s 0 , s 1 ] is affine linear with respect to s 0 and s 1 . The polynomials l 1 , . . . , l 6 correspond to the polynomials with the same name in (3.13), hence by Theorem 3.16 they are the linear relations of the unprojection in the sense of Remark 2.1. The main result of this section is the following theorem, which specifies the quadratic relation of the unprojection ring for the parameter value n = 3. It will be proved in Subsection 4.1. where a was defined in (4.1) and b was defined in (4.2), is the quadratic relation of the type II 1 unprojection for the parameter value n = 3 in the sense of Remark 2.1. In other words, the ideal I Y ⊂ O amb [s 0 , s 1 ] defining, in the sense of Section 2, the type II 1 unprojection is equal to
Linear equations of type II 1 unprojection
where the two polynomials f 1 , f 2 generating I X were defined in (4.3) and the six polynomials l 1 , . . . , l 6 were calculated in (4.4).
4.1
Proof of Theorem 4.1
4.1
We first prove the following reduction lemma.
4.1
Lemma 4.2 To prove Theorem 4.1 it is enough to prove that that the ele-
4.1
Proof Indeed, by [P] 
4.1
Since by the assumptions of the lemma (x 1 + x 2 + x 3 )q ∈ J, and by [P] Section 3.1 J is a prime ideal of O amb [s 0 , s 1 ] we get q ∈ J. Therefore, the affine linear with respect to s 0 , s 1 element q − q is in J, as a consequence (compare proof of [P] Proposition 2.13) we get
4.1
which finishes the proof of Lemma 4.2. QED For the rest of the proof we define some more notation: We define two polynomials g 1 , g 2 , such that, for j ∈ {1, 2}, we have
4.1
where p ∈ {1, 2} is uniquely specified by {j, p} = {1, 2}.
4.1
Moreover, we define two 1 × 3 matrices
4.1
and two 3 × 3 matrices M 1 and M 2 by
4.1
Finally, we define the polynomial
4.1
Proposition 4.3 We have the following equality
4.1
Proof The verification was done using the computer algebra program 
Invariance under SL 3
Let P ∈ SL 3 (Z) be a 3 × 3 matrix with integer coefficients and determinant one. Define two 1 × 3 matrices x = ( x 1 , x 2 , x 3 ) and y = ( y 1 , y 2 , y 3 ) by
Invariance under SL 3
(where P −t is, of course, the transpose matrix of the inverse of P ), two 3 × 3 skew-symmetric matrices A j and two 3 × 3 symmetric matrices B j , for j = 1, 2, by
Invariance under SL 3
and finally set z = z. Moreover, we denote by A j lm the lm-th entry of the matrix A j and by B j lm the lm-th entry of the matrix B j . It can be checked that if we denote by a and b the polynomials of Expressions (4.1) and (4.2) respectively, with the variables without tilde replaced by the corresponding expressions with tilde, we have a = a and b = b.
4.3
How we got to Theorem 4.1
4.3
In this subsection we briefly sketch the method that allowed us to calculate, using the computer program Maple, the expressions for a and b appearing in Theorem 4.1.
4.3
Step 1. Recall the explicit expressions for l i in (4.4). We started from the expression 2 [−s
4.3
which is equal to 2(x 1 + x 2 + x 3 )(s 
4.3
It is easy to see that 2H is a homogeneous quadratic polynomial with respect to the variables A t ij , B s kl . More precisely, 2H has a (unique) natural representation as a sum of terms of the form Step 2. We first handle the 'subpart' of 2H which is sum of terms of the form A 1 ij A 2 kl c A ijkl . These term appear only as contributions from s 0 (l 1 +l 2 +l 3 ) and, moreover, this subpart is equal to 2as 0 , where a was defined in (4.1).
4.3
Step 3. We now handle the 'subpart' of 2H which is sum of terms of the form A 1 ij B 2 kl c AB ijkl . We notice that there is a kind of complementarity of terms for the pairs (l 1 , l 4 ), (l 2 , l 5 ), (l 3 , l 6 ) in the sense of the following example:
4.3
A partial sum of the coefficient of A 1 13 B 2 12 coming from l 1 and l 4 is 2y 1 x 1 s 0 + 2x 2 1 s 1 which can be written as 2x 1 (y 1 s 0 + x 1 s 1 ). Using l 4 we get an expression not involving s 0 and s 1 . Similarly, another partial sum coming from l 2 and l 5 is 2x 1 (y 2 s 0 + x 2 s 1 ) and we can use l 5 to to get an expression not involving s 0 and s 1 . A similar procedure can be done for the the partial sum coming from l 3 and l 6 .
4.3
By symmetry, the handling of the 'subpart' of 2H which is sum of terms of the form A 2 ij B 1 kl c BA ijkl is similar. The handling of the 'subpart' of 2H which is sum of terms of the form B 1 ij B 2 kl c B ijkl is similar, but we need to take care of preserving the symmetries. This is the only part where we actually need the coefficient 2 in 2H.
4.3
To give an example, a partial sum of the coefficient of B 1 13 B 2 23 coming from l 1 and l 4 is 2 [−s 0 (−x 1 x 3 z − y 1 y 3 ) + s 1 (y 3 x 1 + y 1 x 3 )] which we first write as y 3 (y 1 s 0 + x 1 s 1 ) + x 1 (zx 3 s 0 + y 3 s 1 ) + y 1 (y 3 s 0 + x 3 s 1 ) + x 3 (zx 1 s 0 + y 1 s 1 ) and then we use the appropriate linear equations l i to get an expression not involving s 0 and s 1 .
4.3
Step 4. At the end of step 3 we arrived to an expression not involving s 0 , s 1 . By suitable subtraction (in a symmetric way) of multiples of f 1 and f 2 we get a polynomial divisible by x 1 + x 2 + x 3 . The quotient is b, which after some further term by term effort can be written in the form (4.2).
Type II 1 unprojection for n = 2
This subsection contains the explicit equations obtained by Miles Reid for the type II 1 unprojection with parameter value n = 2. It is taken from [R] Section 9.5.
Type II 1 unprojection for n = 2
For n = 2 the ambient ring is corresponding to (2.4) , the linear equations l 1 , . . . , l 4 are given by
Type II 1 unprojection for n = 2
and the quadratic equation q is given by
Type II 1 unprojection for n = 2
In other words, using the notations of Section 2 we have the equality of
Type II 1 unprojection for n = 2
we even get
Type II 1 unprojection for n = 2
Remark 4.5 By [P] the ideal I Y is Gorenstein codimension 3. It is easy to see that it is equal to the ideal generated by the 5 submaximal Pfaffians of the 5 × 5 skew-symmetric matrix (with entries in
Type II 1 unprojection for n = 2
(For a discussion about the Pfaffians of a skew-symmetric matrix see, for example, [BH] Section 3.4.)
Applications to algebraic geometry
We believe that the explicit formulas of Theorem 4.1 can be used together with Gavin Brown's online database of graded rings [Br] for the proof of the existence of a number of (singular) Fano 3-folds with anticanonical ring of codimension 4. We discuss below two such examples, the first of which is due to Selma Altınok, and has the interesting property |− K X | = ∅. We also expect that the explicit formulas of Theorem 4.1 together with the orbifold Riemann-Roch theorem obtained in [BS] can lead to the construction of new codimension 4 Calabi-Yau 3-folds.
The example of Altınok
This example is taken from [R] Example 9.14 and is due to Altınok. Consider the weighted projective space P 2 (1, 3, 5) with coordinates u, v, w and the weighted projective space P 5 (2, 3, 4, 5, 6, 7) with coordinates x, v, y, w, z, t.
The example of Altınok
We define the map P(1, 3, 5) → P (2, 3, 4, 5, 6, 7) given by
The example of Altınok
The equations of the image D of the map are rank y z t xv xw x 4 v w x 3 y z t ≤ 1, that is, yw = vz, yx 3 = vt, zx 3 = wt, y 2 = v 2 x, yz = vwx yt = vx 4 , z 2 = w 2 x, zt = wx 4 , t 2 = x 7 .
The example of Altınok
A general complete intersection X 12,14 containing D is obtained by choosing two general combinations, one of degree 12 and the other of degree 14, of the above equations of D.
The example of Altınok
We perform a type II 1 unprojection of the pair D ⊂ X 12,14 to get a codimension 4 3-fold 2, 3, 4, 5, 6, 7, 8, 9) .
The example of Altınok
Notice that the new variable s 0 has degree 8 and the new variable s 1 has degree 9. After substituting to the formulas (3.13) and (4.5) we checked the quasismoothness of Y using the computer algebra program Singular [GPS01] .
The second Fano example
In this subsection we sketch a construction, suggested by Brown's online database of graded rings [Br] , of a codimension 4 Fano 3-fold
The second Fano example
starting from a (nongeneric) codimension 2 complete intersection Fano 3-fold X 4,6 ⊂ P 5 (1, 1, 2, 2, 2, 3). Write x 1 , x 2 , y 1 , y 2 , y 3 , w for the coordinates of P 5 (1, 1, 2, 2, 2, 3), and consider the subscheme D ⊂ P 5 (1, 1, 2, 2, 2, 3) with equations rank y 1 y 2 w y 3 x 1 y 3 x 2 y 2 3
The second Fano example
x 1 x 2 y 3 y 1 y 2 w ≤ 1.
The second Fano example
By definition, the ideal I D of D is generated by the 2 × 2 minors of the above matrix. Denote by X 4,6 ⊂ P 5 (1, 1, 2, 2, 2, 3) a general codimension 2 complete intersection (with equations of degrees 4 and 6) containing D. The equations of X 4,6 are obtained by choosing two general combinations, one of degree 4 and the other of degree 6, of the above equations of D.
The second Fano example
We perform a type II 1 unprojection of the pair D ⊂ X 4,6 to get a codimension 4 3-fold Y ⊂ P 5 (1, 1, 2, 2, 2, 2, 3, 3). Notice that here the new variable s 0 has degree 2 and the new variable s 1 has degree 3.
The second Fano example
After substituting to the explicit formulas (3.13) and (4.5) we checked the quasismoothness of Y using the computer algebra program Singular [GPS01] .
Appendix: Macaulay 2 code
This section contains the Macaulay 2 code for the type II 1 unprojection with n = 3.
Appendix: Macaulay 2 code
--M2 code for type II_1 unprojection for n=3 kk = QQ S = kk [A1p12, A1p13,A1p23,A2p12,A2p13,A2p23, B1p11,B1p12,B1p13, B1p22,B1p23,B1p33, B2p11,B2p12,B2p13,B2p22,B2p23,B2p33, y1,y2,y3,x1,x2,x3,z,s0, --wedge2: input: single 3x2 matrices u, --output: single 3x1 matrix \wedge^2 (u) wedge2 = (u) -> matrix {{det(submatrix (u,{1,2},{0,1}))}, {-det(submatrix (u,{0,2},{0,1}))}, {det(submatrix (u,{0,1},{0,1}))}} --mixedwedge: input two 3x2 matrices u1 and u2, output --single 3x1 matrix u1 \wedge u2 + u2 \wedge u1 mixedwedge = (u1,u2) -> matrix{ {det (submatrix(u1,{1},{0,1}) || submatrix(u2,{2},{0,1}))+ det (submatrix(u2,{1},{0,1}) || submatrix(u1,{2},{0,1}))}, {-(det (submatrix(u1,{0},{0,1}) || submatrix(u2,{2},{0,1}))+ det (submatrix(u2,{0},{0,1}) || submatrix(u1,{2},{0,1})))}, {det (submatrix(u1,{0},{0,1}) || submatrix(u2,{1},{0,1}))+ det (submatrix(u2,{0},{0,1}) || submatrix(u1,{1},{0,1}))}} x = matrix {{x1,x2,x3}} y = matrix {{y1,y2,y3}} A1 = matrix {{0,A1p12,A1p13},{-A1p12,0,A1p23},{-A1p13,-A1p23,0}} A2 = matrix {{0,A2p12,A2p13},{-A2p12,0,A2p23},{-A2p13,-A2p23,0}} B1 = matrix {{B1p11,B1p12,B1p13},{B1p12,B1p22,B1p23}, {B1p13,B1p23,B1p33}} B2 = matrix {{B2p11,B2p12,B2p13},{B2p12,B2p22,B2p23}, {B2p13,B2p23,B2p33}}
Appendix: Macaulay 2 code
--A1,A2 are generic 3x3 skew, B1,B2 are generic 3x3 symmetric ID = minors (2, matrix {{ y1,y2,y3, z*x1,z*x2, z*x3}, { x1,x2,x3, y1,y2, y3}}) adjB1 = adjointMatrix (B1); adjB1p11 = adjB1_0_0; adjB1p22 = adjB1_1_1; adjB1p33 = adjB1_2_2; adjB1p12 = adjB1_1_0; adjB1p13 = adjB1_2_0; adjB1p23 = adjB1_2_1; adjB2 = adjointMatrix (B2); adjB2p11 = adjB2_0_0; adjB2p22 = adjB2_1_1; adjB2p33 = adjB2_2_2; adjB2p12 = adjB2_1_0; adjB2p13 = adjB2_2_0; adjB2p23 = adjB2_2_1; a = det matrix {{x1,x2,x3},{A1p23,-A1p13,A1p12}, {A2p23,-A2p13,A2p12}}; invof2 = substitute ((1_S/2_S), S); b = (invof2 * (y* B1* adjB2*B1*transpose(y)+ y* B2* adjB1*B2*transpose(y)+ 2*x*A1* adjB2*(transpose A1)*transpose(x)+ 2*x*A2* adjB1*(transpose A2)*transpose(x)-2*x*A1*doubleAdjointMatrix (B1,B2)* transpose(A2)*transpose(x)-z*x*B1*adjB2*B1*transpose(x)-z*x*B2*adjB1*B2*transpose(x)+ 4*y*B1*adjB2*transpose(A1)*transpose(x)+ 4*y*B2*adjB1*transpose(A2)*transpose(x)-(A1p12*(x1*y2-x2*y1)+A1p13*(x1*y3-x3*y1)+

{'paper_id': '15096922', 'title': 'Some calculations on type II_1 unprojection', 's2_url': 'https://api.semanticscholar.org/CorpusID:15096922', 'source': 'S2ORC'}