aa r X i v : . [ m a t h . AG ] S e p Factorial hypersurfaces
A.V.Pukhlikov
In this paper the codimension of the complement to the setof factorial hypersurfaces of degree d in P N is estimated for d > , N > .Bibliography: 12 titles. Key words: factoriality, hypersurface, singularity.
Introduction
A singular point o of an algebraic variety V is factorial , if every prime divisor D ∋ o in a neighborhood of this point is given bya single equation f ∈ O o,V . (For a non-singular point o ∈ V this is a well knowntheorem of the classical algebraic geometry.) Majority of the modern techniqueswork for factorial or Q -factorial varieties (in the latter case it is required that somemultiple of every prime divisor D ∋ o were given by one equation, see, for instance,[1] or any paper on the minimal model program). A standard example of a non- Q -factorial (and the more so, non-factorial) variety is the three-dimensional cone in P over a non-singular quadric in P : no multiple of a plane passing through thevertex of the cone (and contained in that cone) can be given by one equation in thelocal ring of the vertex of the cone. The aim of this paper is to estimate from belowthe codimension of the complement to the set of factorial hypersurfaces of degree d in P N , N > , d > . More precisely, let P d,N +1 be the space of homogeneouspolynomials of degree d in the coordinates x , . . . , x N on P N . Let P fact d,N +1 ⊂ P d,N +1 be the subset, consisting of such f ∈ P d,N +1 that the hypersurface { f = 0 } isirreducible, reduced and factorial. Theorem 0.1. (i)
Assume that d N (that is, { f = 0 } is a Fanohypersurface). Then the estimate codim (cid:0) P d,N +1 \ P fact d,N +1 (cid:1) > min (cid:20) (cid:18) d + N − N − (cid:19) − N, (cid:18) d + N − N − (cid:19)(cid:21) holds, and in the case d = N (that is, { f = 0 } is a Fano hypersurface of index one)in the right hand side one can leave only (cid:0) d + N − N − (cid:1) .(ii) Assume that d > N (in particular, the hypersurface { f = 0 } is a variety of eneral type). Then the following estimate holds: codim (cid:0) P d,N +1 \ P fact d,N +1 (cid:1) > (cid:18) d + N − N − (cid:19) + 4 (cid:18) d + N − N − (cid:19) − N − . In fact, we will obtain an estimate for the codimension of the complement to theset P fact d,N +1 for any values d > (Theorem 3.1), just in the Fano case ( d N ) andin the case d > N that estimate can be essentially simplified to the inequalities ofTheorem 0.1. Our proofis based on the famous Grothendieck’s theorem [2] and the technique of estimatingthe codimension of the set of hypersurfaces in P N with a singular set of prescribeddimension, developed in [3, 4]. Grothendieck’s theorem claims that a variety withhypersurface singularities (in fact, with complete intersection singularities) is factorial,if the singular locus has codimension at least 4. For that reason, in order to estimatethe codimension of the complement to P fact d,N +1 , it is sufficient to estimate the codimensionof the subset consisting of polynomials f ∈ P d,N +1 such that the singular locus ofthe hypersurface { f = 0 } has codimension at most 3. This is what we will do in thepresent paper.The paper is organized in the following way. In §1 we compute the codimensionsof two sets of polynomials f ∈ P d,N +1 : such that the hypersurface { f = 0 } hasa linear space of singular points and such that the hypersurface { f = 0 } has asubvariety of singular points which is a hypersurface in a linear space. After thatwe set the general problem of estimating the codimension of the set of polynomials f ∈ P d,N +1 , such that the singular locus of the hypersurface { f = 0 } is of dimensionat least i > , and state the main technical result — Theorem 1.1, solving thisproblem.§2 contains the proof of Theorem 1.1 by means of the technique developed in[3, 4]. In §3 we obtain an estimate of the codimension of the set of non-factorialhypersurfaces, which implies Theorem 0.1. Factoriality of algebraicvarieties is a very old topic in Algebraic Geometry, with lots of papers written on thesubject. We will only point out a few recent papers that demonstrate that the topicis still actively investigated today: [5, 6, 7, 8, 9]. Various technical points relatedto the constructions of this paper were discussed by the author in his talks givenin 2009-2014 at Steklov Mathematical Institute. The author thanks the members ofdivisions of Algebraic Geometry and Algebra and Number Theory for the interest tohis work. The author is also grateful to his colleagues in Algebraic Geometry groupat the University of Liverpool for the creative atmosphere and general support.2
Hypersurfaces with a large singular locus
In this section we consider the problem of estimating the codimension of the set ofpolynomials f , defining hypersurfaces with a large singular locus. As a first examplewe compute the codimension of the set of polynomials f , such that the hypersurface { f = 0 } has a linear subspace of singular points (Subsection 1.1). The next bycomplexity case, when the singular locus is a hypersurface in a linear subspace, ismade in Subsection 1.2. In Subsection 1.3 we give a precise setting of the problemin the general case and state the main result. Let P N be the complex projective space with homogeneous coordinates ( x : x : · · · : x N ) , N > and P d,N +1 = H ( P N , O P N ( d )) the linear space of homogeneous polynomials of degree d in x ∗ . For a polynomial f ∈ P d,N +1 \ { } the set of singular points of the hypersurface { f = 0 } is denotedby the symbol Sing( f ) . Set P ( i ) d,N +1 = { f ∈ P N,d | dim Sing( f ) > i } , where for the identically zero polynomial f ≡ we set Sing(0) = P N . Obviously, thesets P ( i ) d,N +1 are closed and for i > j we have P ( j ) d,N +1 ⊂ P ( i ) d,N +1 .In order work with the sets P ( i ) d,N +1 , it is convenient to represent them as a union ofmore special subsets that take into account more information about the set Sing( f ) ,not only its dimension. For k > i let P ( i,k ) d,N +1 ⊂ P ( i ) d,N +1 be the closure of the set P ( i ) d,N +1 , consisting of polynomials f , such that Sing( f ) contains an irreducible component C of dimension i > , the linear span h C i ofwhich is a k -plane in P N . For instance, P ( i,i ) d,N +1 consists of polynomials f , such that Sing( f ) contains a i -plane in P N . The closure in this case is not needed: the set P ( i,i ) d,N +1 allows the following obvious explicit description. Proposition 1.1.
The following equality holds: codim (cid:16) P ( i,i ) d,N +1 ⊂ P d,N +1 (cid:17) = (cid:18) d + id (cid:19) + ( N − i ) (cid:18) d + i − d − (cid:19) − ( i + 1)( N − i ) . Proof.
For an i -plane P ⊂ P N by the symbol P ( i,i ) d,N +1 ( P ) we denote the closedset of polynomials f , such that P ⊂ Sing( f ) . Fixing P , we may assume that P = { x i +1 = . . . = x N = 0 } , so that the property f ∈ P ( i,i ) d,N +1 ( P ) is defined by the set of identical equalities f | P ≡ ∂f∂x i +1 (cid:12)(cid:12)(cid:12)(cid:12) P ≡ . . . ≡ ∂f∂x N (cid:12)(cid:12)(cid:12)(cid:12) P ≡ .
3t is easy to see that these equalities are equivalent to vanishing the coefficients atthe monomials in x , . . . , x i and the monomials of the form x j x a x a . . . x a i i ,j = i + 1 , . . . , N , which are all distinct. Therefore, the codimension of the closed set P ( i,i ) d,N +1 ( P ) is (cid:18) d + id (cid:19) + ( N − i ) (cid:18) d + i − d − (cid:19) . Taking into account that for a general polynomial f ∈ P ( i,i ) d,N +1 ( P ) the equality Sing( f ) = P holds for some i -plane P , we obtain the claim of the proposition. Let usconsider now the next by complexity example: let us estimate the codimension ofthe set P ( i,i +1) d,N +1 . This set is the closure of the set of polynomials f , such that for some ( i + 1) -plane P ⊂ P N and an irreducible hypersurface C ⊂ P of degree q > wehave the inclusion C ⊂ Sing( f ) . Fixing the linear subspace P , we obtain the closedsubset P ( i,i +1) d,N +1 ( P ) ⊂ P ( i,i +1) d,N +1 , so that P ( i,i +1) d,N +1 = [ P ⊂ P N P ( i,i +1) d,N +1 ( P ) , where the union is taken over all ( i + 1) -planes in P N . By Bertini’s theorem (and theexplicit description of the polynomials f ∈ P ( i,i +1) d,N +1 ( P ) , given below in the proof ofProposition 1.2), for a general polynomial f ∈ P ( i,i +1) d,N +1 there is a unique ( i + 1) -plane P ⊂ P N , such that f ∈ P ( i,i +1) d,N +1 ( P ) , and for that reason the equality codim (cid:16) P ( i,i +1) d,N +1 ⊂ P d,N +1 (cid:17) = codim (cid:16) P ( i,i +1) d,N +1 ( P ) ⊂ P d,N +1 (cid:17) − ( i + 2)( N − i − holds.Now let us fix P : we may assume that P = { x i +2 = . . . = x N = 0 } . It is clear that C ⊂ Sing( f | P ) . If f | P , then C is a multiple component ofthe hypersurface { f | P = 0 } . There are at most (cid:2) d (cid:3) such components and theyare determined by the polynomial f | P . However, there is also another option: P =Sing( f | P ) , that is, f | P ≡ . In that case the subvariety of singularities C ⊂ P isdetermined by the polynomial f , but not by its restriction f | P . In order to take bothoptions into account, let us write P ( i,i +1) d,N +1 ( P ) = P ( i,i +1; i ) d,N +1 ( P ) ∪ P ( i,i +1; i +1) d,N +1 ( P ) , where P ( i,i +1; l ) d,N +1 ( P ) is the closure of the set of polynomials f ∈ P ( i,i +1) d,N +1 ( P ) , such that dim Sing( f | P ) = l. P ( i,i +1) d,N +1 ( P ) is the minimum of the codimensions of thosetwo sets. It is obvious from the explicit formulas for those codimensions that theminimum is attained at the first set. Proposition 1.2. (i)
For d > , d = 6 or for d = 6 , i N − the followingequality holds: codim (cid:16) P ( i,i +1; i ) d,N +1 ( P ) ⊂ P d,N +1 (cid:17) == (cid:18) d + i + 1 i + 1 (cid:19) − (cid:18) d + i − i + 1 (cid:19) − (cid:18) i + 3 i + 1 (cid:19) + ( N − i − (cid:18)(cid:18) d + ii + 1 (cid:19) − (cid:18) d + i − i + 1 (cid:19)(cid:19) . (ii) For d = 6 , i = N − the following equality holds codim (cid:16) P ( N − ,N ; N − ,N +1 ( P ) ⊂ P ,N +1 (cid:17) = (cid:18) N + 66 (cid:19) − (cid:18) N + 33 (cid:19) − . (iii) The following equality holds codim (cid:16) P ( i,i +1; i +1) d,N +1 ( P ) ⊂ P d,N +1 (cid:17) == min (cid:26)(cid:18) d + i + 1 i + 1 (cid:19) − (cid:18) i + 3 i + 1 (cid:19) + ( N − i − (cid:18)(cid:18) d + ii + 1 (cid:19) − (cid:18) d + i − i + 1 (cid:19)(cid:19) , (cid:18) d + i + 1 i + 1 (cid:19) + ( N − i − (cid:18)(cid:18) d + ii + 1 (cid:19) − (cid:19)(cid:27) . Proof.
Let us show the claim (i). For a general polynomial f ∈ P ( i,i +1; i ) d,N +1 ( P ) wehave: f | P , the hypersurface { f | P = 0 } ⊂ P has a multiple component C ofdegree q > , and moreover, ∂f∂x j (cid:12)(cid:12)(cid:12)(cid:12) C ≡ (1)for j = i + 2 , . . . , N . Note that the coefficients of the polynomials f | P and ∂f∂x j (cid:12)(cid:12)(cid:12) P , j = i + 2 , . . . , N , corrrespond to distinct coefficients of the original polynomial f .The requirement that the hypersurface { f | P = 0 } ⊂ P has a double component C of degree q > , gives E q = (cid:18) d + i + 1 i + 1 (cid:19) − (cid:18) d − q + i + 1 i + 1 (cid:19) − (cid:18) q + i + 1 i + 1 (cid:19) independent conditions for the coefficients of the polynomial f | P . Lemma 1.1.
For d = 6 the minimum of the numbers E q , q = 2 , . . . , [ d/ , isattained at q = 2 . Proof.
We have: ( i + 1)!( E q +1 − E q ) == [( d − q + i + 1) . . . ( d − q + 1) − ( d − q + i − . . . ( d − q − −− [( q + i + 2) . . . ( q + 2) − ( q + i + 1) . . . ( q + 1)] , i !( E q +1 − E q ) == (2 d − q + i )( d − q + i − . . . ( d − q + 1) − ( q + i + 1) . . . ( q + 2) . The first product decreases when q is increasing, the second one is increasing. Itis easy to check that for d > the inequality E > E holds. Therefore, for thesequence of integers E q , q = 2 , . . . , [ d/ , there are two options: • either it is increasing, • or it is first increasing ( q = 2 , . . . , q ∗ ) , and then decreasing ( q = q ∗ , . . . , [ d/ .In the first case the claim of the lemma is obvious. In the second case theminimum of the numbers E q is attained either at q = 2 , or at q = [ d/ , andan easy check shows that the minimum corresponds precisely to the value q = 2 .Q.E.D. for the lemma.Now let us fix the polynomial f | P . Since the hypersurface { f | P = 0 } hasfinitely many components, we may assume that the irreducible hypersurface C ofdegree q > is fixed. Now the requirement (1) imposes on the coefficients of thepolynomial ( ∂f /∂x j ) | P precisely (cid:18) d + ii + 1 (cid:19) − (cid:18) d − q + ii + 1 (cid:19) (2)independent conditions, and it is obvious, that the minimum of the last expressionis attained at q = 2 . This completes the proof of the claim (i) (an explicit checkshows that it is true for d = 6 , too, although for d = 6 the claim of Lemma 1.1 isnot true: E < E .) The claim (ii) is shown by explicit simple computations.Let us show the claim (iii). In that case the hypersurface { f = 0 } containsthe entire subspace P . The closed subset of polynomials f , such that f | P ≡ , hascodimension (cid:0) d + i +1 i +1 (cid:1) . Furthermore, either all partial derivatives ∂f∂x j , j = i + 2 . . . . , N ,vanish on P , so that P ⊂ Sing( f ) and this gives an essentially higher codimensionthan what is claimed by (iii), and for that reason this option can be ignored, or ∂f∂x j | P for some j ∈ { i + 2 , . . . , N } . Without loss of generality we may assumethat j = i + 2 . Then all polynomials ∂f /∂x j , j = i + 3 , . . . , N , vanish on one of thecomponents C of the hypersurface (cid:26) ∂f∂x i +2 = 0 (cid:27) ⊂ P , where deg C = q > . Thiscomponent can be assumed to be fixed and this gives for each of the ( N − i − polynomials ∂f /∂x j , j = i + 3 , . . . , N , the new independent conditions, the numberof which is given by the formula (2). Taking into account that the hypersurface { ∂f /∂x i +2 = 0 } is reducible, we finally obtain (cid:18) d + i + 1 i + 1 (cid:19) + ( N − i − (cid:18)(cid:18) d + ii + 1 (cid:19) − (cid:18) d − q + ii + 1 (cid:19)(cid:19) − (cid:18) i + q + 1 i + 1 (cid:19) independent conditions for the coefficients of the polynomial f . Using the samemethod as in the proof of the claim (i), it is easy to show that the minimum of the6ast expression for q = 2 , . . . , d − is attained at one of the end values of q : eitherat q = 2 , or at q = d − . Proof of Proposition 1.2 is complete. Q.E.D. Now let us considerthe general case. By the symbol P ( i,k ; l ) d,N +1 ( P ) for a k -plane P ⊂ P N denote the closureof the set of polynomials f ∈ P d,N +1 such that: • the set Sing( f ) has an irreducible component C ⊂ P of dimension i , andmoreover h C i = P , • the set Sing( f | P ) has an irreducible component B of dimension l > i , andmoreover C ⊂ B .By the symbol P ( i,k ) d,N +1 ( P ) for a k -plane P ⊂ P N denote the closure of the setof polynomials f ∈ P d,N +1 such that the first of the two conditions stated above issatisfied. Obviously, P ( i,k ) d,N +1 ( P ) = k [ l = i P ( i,k ; l ) d,N +1 ( P ) . Everywhere in the sequel the codimension of various closed sets in the space ofpolynomials P d,N +1 is meant to be with respect to that space, so that, for instance, codim P ( i,k ) d,N +1 ( P ) is the minimum of the codimensions codim P ( i,k ; l ) d,N +1 ( P ) , where l = i, . . . , k , and the following estimate holds: codim P ( i,k ) d,N +1 > codim P ( i,k ) d,N +1 ( P ) − ( k + 1)( N − k ) . Remark 1.1.
It is easy to see that for N − k < l − i the singular set of apolynomial f ∈ P ( i,k ; l ) d,N +1 ( P ) is of dimension at least i + 1 . Indeed, Sing( f | P ) containsan l -dimensional irreducible component C ⊂ P , where l > i . If k = N , then C ⊂ Sing( f ) . If k < N , then (cid:20) C ∩ (cid:26) ∂f∂x k +1 (cid:12)(cid:12)(cid:12)(cid:12) P = . . . = ∂f∂x N (cid:12)(cid:12)(cid:12)(cid:12) P = 0 (cid:27)(cid:21) ⊂ Sing( f ) , so that dim Sing( f ) > l − ( N − k ) > i , as we claimed. For that reason everywherebelow, whenever we consider the sets P ( i,k ; l ) d,N +1 ( P ) , we assume that the inequality N + i > k + l holds.In order to give a compact statement of the main result about these codimensions,we introduce one notation more. For positive integers a, b, c set τ ( a, b, c ) = max (cid:26)(cid:18) a + cc (cid:19) , ab + 1 (cid:27) . If we fix c , then the first of the two numbers exceeds the second one for the valuesof a that are higher than a number of order ce N c , where e is the base of the naturallogarithm. If we fix a , the first of the two numbers exceeds the second one for the7alues of c that are higher than a number of order ae N a . The meaning of the function τ will be clear in §2. Theorem 1.1.
For l k − the following estimate holds: codim P ( i,k ; l ) d,N +1 ( P ) > ( k − l + 1) (cid:18) d + l − l + 1 (cid:19) + ( N + i − k − l ) τ ( d − , k, i ) . Theorem 1.2. (i)
For l = k − , d = 6 the following estimate holds: codim P ( i,k ; l ) d,N +1 ( P ) > (cid:20)(cid:18) d + kk (cid:19) − (cid:18) d − kk (cid:19) − (cid:18) k + 2 k (cid:19)(cid:21) +( N + i − k +1) τ ( d − , k, i ) . (ii) For l = k − , d = 6 the following estimate holds: codim P ( i,k ; l ) d,N +1 ( P ) > (cid:20)(cid:18) k + 6 k (cid:19) − (cid:18) k + 3 k (cid:19)(cid:21) + ( N + i − k + 1) τ (5 , k, i ) . Theorem 1.3.
For l = k the following estimate holds: codim P ( i,k ; l ) d,N +1 ( P ) > (cid:18) d + kk (cid:19) + ( N + i − k − l ) τ ( d − , k, i ) . Proof of these three theorems will be given in §2.
In this section we prove Theorems 1.1-1.3. Theorem 1.1 is the hardest one. First(Subsection 2.1) we describe the strategy of the proof of this theorem and give thedefinition of good sequences and associated subvarieties: this technique amkes itpossible to reconstruct the subvariety C ⊂ Sing( f ) inside the, generally speaking,larger subvariety B ⊂ Sing( f | P ) . After that we estimate the codimension of thesubset of polynomials on P with a non-degenerate subvariety of singular points ofdimension l (Subsection 2.2). Finally, in Subsection 2.3 we complete the proof ofTheorem 1.1 by means of well known methods of estimating the codimension of theset of polynomials, vanishing on a given non-degenerate subvariety; after that weshow Theorems 1.2 and 1.3, which is easy. Let us describe thestrategy of the proof of Theorem 1.1. Fix a k -plane P ⊂ P N . We will assume that itis the coordinate plane P = { x k +1 = . . . = x N = 0 } . As we noted in Subsection 1.2, the coefficients of the polynomials f | P and ∂f∂x j (cid:12)(cid:12)(cid:12) P , j = k + 1 , . . . , N , correspond to distinct coefficients of the polynomial f . For thatreason, considering the general polynomial f ∈ P ( i,k ; l ) d,N +1 ( P ) , one has to solve threeproblems: 8) estimate the codimension of the closed subset P ( l,k ) d,k +1 in the space P d,k +1 (since,obviously, f | P ∈ P ( l,k ) d,k +1 ),2) using the l − i polynomials ∂f∂x j (cid:12)(cid:12)(cid:12) P , where j ∈ I ⊂ { k + 1 , . . . , N } , so that | I | = l − i , reconstruct the variety of singular points C ⊂ Sing( f ) as a subvarietyof codimension l − i of the variety of singular points B ⊂ Sing( f | P ) , which dependson the restriction f | P only and for that reason can be assumed to be fixed, if we fixthe polynomial f | P ∈ P ( l,k ) d,k +1 ,3) estimate the codimension of the closed set of polynomials h ∈ P d − ,k +1 ,vanishing on a fixed non-degenerate subvariety B ⊂ P , and apply this estimateto the ( N + i − k − l ) polynomials ∂f∂x j (cid:12)(cid:12)(cid:12) P , j ∈ { k + 1 , . . . , N } , j I .The sum of the estimate, obtained at the stage 1), with the ( N + i − k − l ) -multiple of the estimate, obtained at the stage 3), is precisely the inequality, claimedby Theorem 1.1.Let us start to realize this programme.First of all, recall the following definition (see [12, Section 3] or [11, Chapter 3]). Definition 2.1.
A sequence of homogeneous polynomials g , . . . , g m of arbitrarydegrees on the projective space P e , e > m + 1 , is said to be a good sequence , and anirreducible subvariety W ⊂ P e of codimension m is its associated subvariety , if thereexists a sequence of irreducible subvarieties W j ⊂ P e , codim W j = j (in particular, W = P e ) such that: • g j +1 | W j for j = 0 , . . . , m + 1 , • W j +1 is an irreducible component of the closed algebraic set g j +1 | W j = 0 , • W m = W .A good sequence can have more than one associated subvarieties, but their numberis bounded from above by a constant, depending on the degrees of the polynomials g j only (see [12, Section 3]).Assuming the polynomial f | P and the subvariety B to be fixed, let us constructa good sequence of polynomials on P = P k with the subvariety C as one of itsassociated subvarieties. This sequence starts with g = f | P . Since B is anirreducible l -dimensional component of the closed set Sing( f | P ) , for some ( k − l − polynomials ∂f∂x j (cid:12)(cid:12)(cid:12) P , j ∈ { , . . . , k } , we obtain a good sequence of polynomialswith B as one of its associated subvarieties. If l = i , then there is nothingmoreto construct. Assume that l > i + 1 . Then among the polynomials ∂f∂x j (cid:12)(cid:12)(cid:12) P , j ∈{ k + 1 , . . . , N } , there is one which does not vanish on B (otherwise, B ⊂ Sing( f ) ,so that f ∈ P ( l,k ; l ) d,N +1 ( P ) , and this contradicts to the assumption that C ⊂ B , C = B is an irreducible component of the set Sing( f ) ). We add this polynomial to alreadyconstructed sequence. Continuing in this way (using at every step the fact that C
9s an irreducible component of the set
Sing( f ) ), we complete the construction of agood sequence. Assuming the polynomials of the good sequence to be fixed, we mayassume that the variety C ⊂ P to be fixed as well. This solves the problem 2), statedabove. Now let us consider the most difficult problem 1). The problem 1), stated above,is solved in the following claim.
Proposition 2.1.
The following estimate holds: codim (cid:16) P ( l,k ) d,k +1 ⊂ P d,k +1 (cid:17) > ( k − l + 1) (cid:18) d + l − l + 1 (cid:19) . (3) Proof.
In order to simplify the notations, we assume that k = N . Let us describethe technique of estimating the codimension of the closed subset of the space P d,N +1 ,consisting of polynomials with many singular points. The following claim is true. Lemma 2.1.
Assume that d > . For any set of m linearly independent points p , . . . , p m ∈ P N , m N + 1 , the condition { p , . . . , p m } ⊂ Sing( g ) ,g ∈ P d,N +1 , defines a linear subspace of codimension m ( N + 1 ) in P d,N =1 . Proof.
We may assume that p = (1 : 0 : 0 . . . : 0) , p = (0 : 1 : 0 : . . . : 0) and so on correspond to the first m vectors of the standard basis of the linearspace C N +1 . The condition p i ∈ Sing( g ) means vanishing of the coefficients at themonomials x di − , x d − i − x j , for all j = i − . For d > all these m ( N + 1) monomialsare distinct. Q.E.D. for the lemma.Now let us consider an arbitrary linear subspace Π ⊂ P N of codimension r + 1 ,where r > , given by a system of r + 1 equations l ( x ) = 0 , l ( x ) = 0 , . . . , l r ( x ) = 0 , where l , . . . , l r are linearly independent forms. For each i = 1 , . . . , r fix an arbitraryset of distinct constants λ i , . . . , λ i,d − ∈ C ; we assume that λ i = 0 for all i =1 , . . . , r . Now for any integer valued point e = ( e , . . . , e r ) ∈ Z r + , e i d − , by the symbol Θ( e ) we denote the linear subspace { l i ( x ) − λ i,e i l ( x ) = 0 | i = 1 , . . . , r } ⊂ P N of codimension r . Obviously, Θ( e ) ⊃ Π . Set | e | = e + . . . + e r ∈ Z + . e ∈ Z r + with | e | d − consider an arbitrary set S ( e ) = { p ( e ) , . . . , p m ( e ) } ⊂ Θ( e ) \ Π of m linearly independent points (so that m N − r + 1 ). Proposition 2.2.
The set of conditions S ( e ) ⊂ Sing(g | Θ(e) ) ,e ∈ Z r + , | e | d − , defines a linear subspace of codimension m ( N − r + 1) | ∆ | in P d,N +1 , where ∆ = { e > , . . . , e r > , e + . . . + e r d − } ⊂ R r is an integer valued simplex and | ∆ | is the number of integral points in that simplex, | ∆ | = ♯ (∆ ∩ Z r ) . Proof.
We may assume that l = x , l = x , . . . , l r = x r . In order to simplify theformulas we will prove the affine version of the proposition: set v = x /x , . . . , v r = x r /x and u i = x r + i /x , i = 1 , . . . , N − r . In the affine space A N ⊂ P N , A N = P N \{ x = 0 } with coordinates ( u, v ) = ( u , . . . , u N − r , v , . . . , v r ) the affine spaces A ( e ) = Θ( e ) \ Π are contained entirely: A ( e ) = Θ( e ) ∩ A N , so that S ( e ) ⊂ A ( e ) for all e . Obviously, A ( e ) = { v = λ ,e , . . . , v r = λ r,e r } ⊂ A N is a ( N − r ) -plane, which is parallel to the coordinate ( N − r ) -plane ( u , . . . , u N − r , , . . . , .Now let us write the polynomial g in terms of the affine coordinates ( u, v ) in thefollowing way: g ( u, v ) = X e ∈ Z r + , | e | d g e ,...,e r ( u ) r Y i =1 e i − Y j =0 ( v i − λ ij ) (if e i = 0 , then the corresponding product is meant to be equal to 1). Here g e ( u ) = g e ,...,e r ( u ) is an affine polynomial in u , . . . , u N − r of degree deg g e d − | e | . When λ ij are fixed, this expression is unique. By Lemma 2.1, the condition S (0) = S (0 , . . . , ⊂ Sing( g | A (0) ) defines a linear subspace of codimension m ( N − r + 1) in the space of polynomials P d,N − r +1 . However it is easy to see that g | A (0) = g ,..., ( u ) , e = 0 in the product r Y i =1 e i − Y j =0 ( v i − λ ij ) there is at least one factor ( v i − λ i ) = v i , which vanishes when we restrict it ontothe ( N − r ) -plane A (0) . Therefore, the condition S (0) ⊂ Sing( g | A (0) ) imposes on thecoefficients of the polynomial g ,..., ( u ) precisely m ( N − r +1) independent conditions,whereas the polynomials g e ( u ) for e = 0 can be arbitrary.Now let us complete the proof of Proposition 2.2 by induction on | e | . Moreprecisely, for every a ∈ Z + set ∆ a = { e > , . . . , e r > , e + . . . + e r a } ⊂ R r , so that ∆ = ∆ d − . Let us prove the claim of Proposition 2.2 in the following form:for every a = 0 , . . . , d − ∗ ) a the set of conditions S ( e ) ⊂ Sing( g | Θ( e ) ) ,e ∈ Z r + , | e | a , defines a linear subspace of codimension m ( N − r +1) | ∆ a | in P d,N +1 , and, moreover, the linear conditionsare imposed on the coefficients of the polynomials g e ( u ) for e ∈ ∆ a , whereas for e ∆ a the polynomials g e ( u ) can bearbitrary.The case a = 0 has already been considered, so that assume that a d − and the claims ( ∗ ) j for j = 0 , . . . , a have been shown. Let us show the claim ( ∗ ) a +1 .Let e ∈ Z r + be an arbitrary multi-index, | e | = a + 1 . The restriction onto theaffine subspace A ( e ) means the substitution v = λ ,e , . . . , v r = λ r,e r . Therefore thepolynomial g e ( u ) comes into the restriction g | A ( e ) with the non-zero coefficient α e = r Y i =1 e i − Y j =0 ( λ i,e i − λ ij ) . On the other hand, for e ′ = e , | e ′ | > a + 1 the product r Y i =1 e ′ i − Y j =0 ( λ i,e i − λ ij ) . is equal to zero, since for at least one index i ∈ { , . . . , r } we have e ′ i > e i andso this product contains a factor equal to zero. Therefore, g | A ( e ) is the sum of thepolynomial α e g e and a linear combination of the polynomials g e ′ with | e ′ | a withconstant coefficients. Now, fixing the polynomials g e ′ with | e ′ | a , we obtain thatthe condition S ( e ) ⊂ Sing( g | A ( e ) ) affine (generally speaking, not linear) subspace of codimension m ( N − r + 1) of the space of polynomials g e ( u , . . . , u N − r ) of degree at most d − | e | , thecorresponding linear subspace of which is given by the condition S ( e ) ⊂ Sing g e ( u ) . Moreover, no restrictions are imposed on the coefficients of other polynomials g e ′ with | e ′ | = a + 1 .This proves the claim ( ∗ ) a for all a = 0 , . . . , d − . Proof of Proposition 2.2 iscomplete.Now let Θ = Θ[ l , . . . , l r ; λ i,j , i = 1 , . . . , r, j = 0 , . . . , d −
1] = { Θ( e ) | e ∈ ∆ } be some set of linear subspaces of codimension r in P N , considered in Proposition2.2. We define the subset P d,N +1 (Θ) ⊂ P d,N +1 by the following condition: for every subspace Θ( e ) with | e | d − there is aset S ( e ) ⊂ Θ( e ) \ Π , consisting of m linearly independent points, such that S ( e ) ⊂ Sing( g | Θ( e ) ) . Proposition 2.3.
The following inequality holds: codim( P d,N +1 (Θ) ⊂ P d,N +1 ) > m | ∆ | . Proof is obtained by means of the obvious dimension count: the subspaces Θ( e ) are fixed, so that every point p i ( e ) varies in a ( N − r ) -dimensional family. Q.E.D.for the proposition.Finally, let us complete the proof of Proposition 2.1. Set N = k , so that thespace of polynomials of degree d is P d,k +1 . For an arbitrary set Θ = { Θ( e ) | e ∈ ∆ } of linear subspaces of codimension l in P = P k let P ( l,k ) d,k +1 ( P, Θ) ⊂ P ( l,k ) d,k +1 be the set of polynomials h ∈ P ( l,k ) d,k +1 such that the set Sing( h ) has an irreduciblecomponent Q of dimension l , where h Q i = P and the variety Q is in general positionwith the subspaces from the set Θ : for all e ∈ ∆ the set Θ( e ) ∩ Q contains ( k − l + 1) linearly independent points. Since h Q i = P , the subset P ( l,k ) d,k +1 ( P, Θ) is a Zariskiopen subset of the set P ( l,k ) d,k +1 , so that the inequality (3) will be shown, if we proveit for P ( l,k ) d,k +1 (Θ) instead of P ( l,k ) d,k +1 . However, for P ( l,k ) d,k +1 (Θ) this inequality followsimmediately from Proposition 2.3, since in that case m = k − l +1 and | ∆ | = (cid:0) d + l − l +1 (cid:1) .Proof of Proposition 2.1 is complete. Q.E.D. By the symbol P d − ,k +1 ( B ) we denote the closed subset of polynomials h ∈ P d − ,k +1 such that h | B ≡ for a13xed irreducible subvariety B . In our case dim B = i and h B i = P k . There are twomethods of estimating the codimension of the set P d − ,k +1 ( B ) .The first method was developed in [10] (see also [11, Chapter 3]). Consider ageneral linear projection P k P i , so that π | B is a regular surjective map. For anynon-zero polynomial g ∈ P d − ,i +1 we have π ∗ g | B , so that codim P d − ,k +1 ( B ) > P d − ,i +1 = (cid:18) d − ii (cid:19) . The second method was developed in [12] (see also [11, Chapter 3]). Since h B i = P k , a non-zero linear form can not vanish on B . Therefore, the closed subset ofdecomposable forms P ,k +1 · P ,k +1 · · · · · P ,k +1 | {z } d − ⊂ P d − ,k +1 intersects with P d − ,k +1 ( B ) by zero only, so that codim P d − ,k +1 ( B ) > ( d − k + 1 . Finally we get: codim( P d − ,k +1 ( B ) ⊂ P d − ,k +1 ) > τ ( d − , k, i ) . By the arguments of Subsection 2.1 about good sequences and Proposition 2.1, thiscompletes the proof of Theorem 1.1.Let us show Theorems 1.2 and 1.3. By the arguments given above, we only needto prove the inequalities codim( P ( k − ,k ) d,k +1 ⊂ P d,k +1 ) > (cid:18) d + kk (cid:19) − (cid:18) d − kk (cid:19) − (cid:18) k + 2 k (cid:19) for d = 6 and codim( P ( k − ,k )6 ,k +1 ⊂ P ,k +1 ) > (cid:18) k + 6 k (cid:19) − (cid:18) k + 3 k (cid:19) (both are in fact equalities), which are obtained by repeating the arguments thatwere used in the proof of Proposition 1.2 word for word. The claim of Theorem 1.3is obvious. In this section we prove Theorem 0.1. First we give the list of possible values of theparameters i = N − , k, l , so that the complement to the set of polynomials P fact d,N +1
14s contained in the union of the sets P ( i,k ; l ) d,N +1 . After that, the estimates obtainedin §§1-2 are applied in order to estimate the codimension of the set of non-factorialhypersurfaces (Subsection 3.1); this estimate is used for proving part (ii) of Theorem0.1 in Subsection 3.2 and part (i) of Theorem 0.1 in Subsection 3.3. Notethe following simple
Proposition 3.1.
For N > , d > the following equality holds: τ ( d − , k, N −
4) = (cid:18) d + N − N − (cid:19) . Proof: obvious computations.By Grothendieck’s theorem, P d,N +1 \ P fact d,N +1 ⊂ P ( N − d,N +1 , so that in order to estimate the codimension of the set of non-factorial hypersurfaces,we will estimate the codimension of the set of hypersurfaces with the singular locusof codimension three. In the notations of §1, the set P ( N − d,N +1 is the union of thefollowing eight sets: P ( N − ,N − d,N +1 , P ( N − ,N − d,N +1 , P ( N − ,N − N − d,N +1 , P ( N − ,N − N − d,N +1 , P ( N − ,N − N − d,N +1 , P ( N − ,N − N − d,N +1 , P ( N − ,N − N − d,N +1 , P ( N − ,N ) d,N +1 . We set respectively: α = (cid:18) d + N − N − (cid:19) + 4 (cid:18) d + N − N − (cid:19) − N − ,α = min { α a , α b } , where α a = (cid:18) d + N − N − (cid:19) − (cid:18) d + N − N − (cid:19) +3 (cid:20)(cid:18) d + N − N − (cid:19) − (cid:18) d + N − N − (cid:19)(cid:21) − ( N + 5)( N − ,α b = (cid:18) d + N − N − (cid:19) + 2 (cid:18) d + N − N − (cid:19) − N − . Furthermore, α = 3 (cid:18) d + N − N − (cid:19) + 2 (cid:18) d + N − N − (cid:19) − N − ,α = (cid:18) d + N − N − (cid:19) − (cid:18) d + N − N − (cid:19) + (cid:18) d + N − N − (cid:19) − ( N + 4)( N − and α ′ = (cid:18) N + 46 (cid:19) − (cid:18) N + 13 (cid:19) + (cid:18) N − (cid:19) − N − N only. Finally, α = (cid:18) d + N − N − (cid:19) − N − ,α = 4 (cid:18) d + N − N − (cid:19) + (cid:18) d + N − N − (cid:19) − N,α = 3 (cid:18) d + N − N − (cid:19) − N and α = 5 (cid:18) d + N − N − (cid:19) . Now Propositions 1.1, 1.2 and Theorems 1.1-1.3, taking into account Proposition3.1, immediately imply
Theorem 3.1.
For N > the following inequality holds: codim (cid:0) P d,N +1 \ P fact d,N +1 (cid:1) > min { α i | i = 1 , . . . , } . Remark 3.1.
For d = 6 in this inequality one should replace α by α ′ , however,the minimum of the right hand side is attained at α all the same, so that the claimof Theorem 3.1 remains correct in this case as well. In oredr to prove the claim (ii) ofTheorem 0.1, one needs to check that α i > α for i > , if d > N . This checkis elementary and we do not perform it here, giving only one example: setting d = 2 N + a , write α − α = 4( N −
3) + (3 N + a − N − N + a )! ×× [10 N (2 N + a − N + a − − ( N − N + a − N + 5 a − . It is easy to see that for a > the expression in the square brackets is positive, whichimplies that α > α . (In fact, the difference α − α is quite large, but we do notneed that.) The remaining inequalities α i > α for i = 2 are shown in a similar way.The reason why α realizes the minimum of α i , i = 1 , . . . , , is the polynomialityof the functions α i in d when the dimension N is fixed: α is a polynomial of theleast degree N − . The polynomial α a also has the degree N − , but its seniorcoefficient is much higher. The claim (ii) of Theorem 0.1 is shown.We see that for d > N the irreducible component of the maximal dimension ofthe closed set P ( N − d,N +1 is P ( N − ,N − d,N +1 , that is, the set of polynomials f ∈ P d,N +1 suchthat the hypersurface { f = 0 } has a ( N − -lane of singular points. Let us prove the claim (i) of Theorem 0.1. Again anelementary (but tiresome) check shows that for i = 1 , . . . , the inequality α i > min { α , α } d = 6 with α replaced by α ′ ), which implies the claim (i). We do notgive the tiresome computations here, except for one example: α − α = − N + ( d + N − N − d − − d + d ( N −
1) + ( N − N + 18)] . It is easy to see that for d = 4 , . . . , N the difference α − α is positive. In a similarway the other inequalities α i > α for i = 1 , . . . , are checked. Proof of the claim(i) of Theorem 0.1 is complete. Remark 3.2.
Elementary computations, which we do not give here, show thatfor d = 4 , . . . , d ∗ ( N ) the inequality α α holds, and for d = d ∗ ( N ) + 1 , . . . , N theopposite inequality α > α holds. Here d ∗ ( N ) ∼ N . More precisely, if N = 3 m + e , e ∈ { , , } , then d ∗ ( N ) = 2 m + e + 1 . Список литературы [1] Mella M., Birational geometry of quartic 3-folds. II. The importance of being Q -factorial. Math. Ann. (2004), No. 1, 107-126.[2] Call F. and Lyubeznik G., A simple proof of Grothendieck’s theorem on theparafactoriality of local rings, Contemp. Math. (1994), 15-18.[3] A.V.Pukhlikov, Birational geometry of Fano hypersurfaces of index two. Math.Ann. Published online 26 December 2015. DOI 10.1007/s00208-015-1345-2.[4] A.V.Pukhlikov, Birational geometry of algebraic varieties fibred into Fanodouble spaces. Izvestiya: Mathematics (in press). arXiv:1512.05681.[5] I.Cheltsov, The factoriality of nodal threefolds and he connectedness of theset of log canonical singularities. Sbornik: Mathematics. (2006), No. 3-4,387-414.[6] I.Cheltsov, Factorial threefold hypersurfaces, J. Alg. Geom. (2010), No. 4,781-791.[7] I.Cheltsov and J.Park, Factorial hypersurfaces in P with nodes. Geom.Dedicata (2006), 205-219.[8] C.Ciliberto and V. Di Gennaro, Factoriality of certain threefold completeintersections in P with ordinary double points. Comm. Algebra (2004),No. 7, 2705-2710.[9] F.Polizzi, A.Rapagnetta and P.Sabatino, On factoriality of threefolds withisolated singularities, Michigan Math. J. (2014), No. 4, 781-801.[10] Pukhlikov A. V., Birational automorphisms of Fano hypersurfaces, Invent.Math. (1998), no. 2, 401-426. 1711] Pukhlikov Aleksandr, Birationally Rigid Varieties. Mathematical Surveys andMonographs , AMS, 2013.[12] Pukhlikov A. V., Birationally rigid Fano complete intersections, Crelle J. f¨urdie reine und angew. Math. (2001), 55-79.Department of Mathematical Sciences,The University of Liverpool [email protected]@liv.ac.uk