Developable cubics in P 4 and the Lefschetz locus in GOR(1,5,5,1)
aa r X i v : . [ m a t h . AG ] D ec DEVELOPABLE CUBICS IN P AND THE LEFSCHETZLOCUS IN
GOR(1 , , , THIAGO FASSARELLA, VIVIANA FERRER, AND RODRIGO GONDIM
Abstract.
We provide a classification of developable cubic hypersur-faces in P . Using the correspondence between forms of degree 3 on P and Artinian Gorenstein K -algebras, given by Macaulay-Matlis duality,we describe the locus in GOR(1 , , ,
1) corresponding to those algebraswhich satisfy the Strong Lefschetz property. Introduction
We work over an algebraically closed field K of characteristic zero. Theprojective space over K of dimension N will be denoted by P N . In thisnote we will focus our attention on the classification of developable cubichypersurfaces on P as well as the Artinian Gorenstein algebras defined bythem.An irreducible projective variety X ⊂ P N is called developable if it has adegenerate Gauss map. Recent progress on the classification problem of de-velopable varieties has been made via the focal locus of the ruling defined byfibers of the Gauss map. For instance, see [AG, MT] for a classification ofdevelopable threefolds. In Section 2 we proceed with a careful analysis of thefocal locus to provide a finer classification of developable cubic hypersurfacesin P . Our first goal is the following result. Theorem 1.1.
Let X ⊂ P be an irreducible cubic hypersurface. Assumethat X is not a cone. Then X is developable if and only if it is projectivelyequivalent to a linear section of the secant variety of the Veronese surface. A linear section of the secant variety of the Veronese surface is projectivelyequivalent to one of the following varieties (see Section 2.4):(1) the secant variety of the rational normal quartic curve;(2) the join of two irreducible conics sharing a single point; this pointcoincides with the intersection between the planes containing theconics.(3) the dual variety of the scroll surface S (1 , Cases (1) and (2) correspond to cubic hypersurfaces which have nonvanishingHessian while case (3) yields a cubic with vanishing Hessian.To each cubic hypersurface above we can associate an Artinian Gorenstein K -algebra. More generally, Macaulay-Matlis duality offers a correspondencebetween forms of degree d in N + 1 variables over K , not defining a cone,and standard graded Artinian Gorenstein K -algebras of socle degree d andcodimension N + 1. These algebras enjoy nice properties such as Poincar´eduality in cohomology theory. We give a precise definition in Section 3.Given a standard graded Artinian Gorenstein K -algebra A = d M k =0 A k , AGalgebra for short, its Hilbert vector is the vector Hilb( A ) = (1 , a , . . . , a d )where a k = dim K A k . We denote by GOR( T ) the space which parametrizesAG algebras with Hilbert vector T . It has been extensively studied in [IK].We are interested in algebras inside GOR( T ) which satisfy the Strong Lef-schetz property, SLP for short, which means that there exists a linear form l ∈ A such that every multiplication map µ l j : A k → A k + j has maximalrank. This notion was introduced in the commutative algebra setting by R.Stanley and J. Watanabe, see [St, W, HW], and was inspired by the so calledhard Lefschetz Theorem on the cohomology of smooth projective complexvarieties, see for example [GH]. The Lefschetz properties have attracted alot of attention over the last years; we refer to [HW] for a survey on thearea.Now we go back to cubic hypersurfaces in P which are not cones. Theycorrespond to AG algebras with Hilbert vector (1 , , , , , ,
1) which come from a developable cubic hypersurface. Algebrasassociated to cases (1) and (2) above have the SLP, whereas it fails in case(3). Moreover, by the main result of [MW], any AG algebra with Hilbertvector (1 , , ,
1) failing SLP comes from case (3). Sections 3 and 4 aredevoted to the description of the locus of algebras in GOR(1 , , ,
1) failingthe SLP. The main results of these sections are summarized by the followingtheorem.
Theorem 1.2.
The space
GOR(1 , , , parametrizing AG algebras withHilbert vector (1 , , , coincides with P \C , where C is the space of cubiccones in P . Moreover, the following assertions hold: (1) The locus C is the image of a projective bundle over P by a bira-tional morphism, its dimension is and its degree is . (2) The locus of algebras failing SLP coincides with
K \ C , where K is arational projective variety of dimension and degree . Moreprecisely, K is the image of a projective bundle over the Grassman-nian G (2 , by a birational morphism. (3) The intersection
K ∩ C is a divisor in K of degree . EVELOPABLE CUBICS IN P AND THE LEFSCHETZ LOCUS IN GOR(1 , , ,
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We note that the locus of algebras in GOR(1 , , ,
1) failing SLP coincideswith the locus of algebras with Jordan type 4 ⊕ ⊕ , while any otheralgebra has Jordan type 4 ⊕ . In particular 4 ⊕ ⊕ is the only possibledegeneration of the general Jordan type. This phenomenon cannot occur for N ≤
3: in this situation having vanishing Hessian is equivalent to be a cone.Hence, for N ≤
3, any algebra in GOR(1 , N + 1 , N + 1 ,
1) has the SLP, see[CG]. 2.
Developable cubics in P Basic definitions.
Given a rational map ϕ : X Y between projec-tive varieties, its image is the closure of ϕ ( U ) in Y , where U is the maximaldomain where ϕ is defined. We say that ϕ : X Y is dominant if it has Y as image.Let X ⊂ P N be a projective subvariety of dimension n ≥
1. Let ( P N ) ∗ denote the space of hyperplanes in P N . We denote by Con X ⊂ P N × ( P N ) ∗ the conormal variety of X : this is the closure of the set of pairs ( x, H ) suchthat x is a regular point of X and H contains the tangent space T x X . Let X ∗ be the image of the projection in the second coordinate. It is the dualvariety of X . Given a point x ∈ P N , we define x ∗ ⊂ ( P N ) ∗ as the set ofhyperplanes passing through it.Let G ( n, N ) denote the Grassmannian of n -planes in P N . The Gaussmap γ : X G ( n, N ) associates to each regular point x ∈ X the tangentspace T x X ∈ G ( n, N ). We denote by X ∨ the image of γ . We say that anirreducible projective variety X is developable if dim X ∨ < n. We are particularly interested in the case where X is a hypersurface.Assume that it is the zero locus X = V ( f ) of a non-constant homogenouspolynomial f in N + 1 variables. Its polar map is the rational mapΦ f : P N ( P N ) ∗ p ( f ( p ) : f ( p ) : ... : f N ( p ))where f i is the partial derivative of f with respect to x i . We denote by Z theimage of the polar map, called polar image of X . The restriction of the polarmap to X is just the Gauss map γ : X ( P N ) ∗ , and X ∨ coincides with X ∗ . We note that since we are working in characteristic zero, the ReflexivityTheorem says that ( X ∗ ) ∗ = X , see [H, p. 208] for an elementary proof.Let us denote by Hess f the Hessian matrix of f , namely the matrix of thesecond derivatives. Its determinant is the Hessian determinant . We shallsay that X = V ( f ) or f has vanishing Hessian , if its Hessian determinantis null. Therefore Φ f is nondominant if and only if f has vanishing Hessian.This is equivalent to say that the derivatives f , . . . , f N of f are algebraicallydependent. We summarize the above discussion in the following proposition. Proposition 2.1.
Let X = V ( f ) ⊂ P N be a hypersurface and Z its polarimage. The following conditions are equivalent. (1) X has vanishing Hessian; FASSARELLA, FERRER, AND GONDIM (2)
The partial derivatives of f are algebraically dependent; (3) Z is a proper subvariety of ( P N ) ∗ . The singular locus and the polar image of a hypersurface with vanishingHessian have a relevant role. The following proposition gives a relationbetween them. Its proof can be found in the original work of Perazzo, see[P] for the cubic case and [Za2, p. 21] for any degree.
Proposition 2.2 ([P, Za2]) . Let X ⊂ P N be a hypersurface with vanishingHessian. Then Z ∗ ⊂ Sing( X ) . Remark 2.3.
Hypersurfaces with vanishing Hessian are developable. Tosee this, we assume that X has vanishing Hessian. First we note that X ∗ is a proper subvariety of Z . In fact if X ∗ = Z then Z ∗ = X , but thiscontradicts Proposition 2.2. The strict inclusions of irreducible varieties X ∗ ( Z ( ( P N ) ∗ imply that dim X ∗ < N −
1, hence X is developable.Given projective subvarieties V, W ⊂ P N , we denote by S ( V, W ) the join between them. It is the closure of the union of lines in P N joining V to W . Inparticular S ( V ) = S ( V, V ) is the secant variety of V . A subvariety V ⊂ P N is a cone if there exists x ∈ V such that S ( x, V ) = V . This motivates thedefinition of the vertex of V Vert( V ) = { x ∈ V : S ( x, V ) = V } . Cones are the simplest examples of hypersurfaces with vanishing Hessian.Now we state the following useful proposition the proof of which will be leftto the reader.
Proposition 2.4.
Let X = V ( f ) ⊂ P N be a hypersurface. Then the follow-ing conditions are equivalent: (i) X is a cone; (ii) The partial derivatives of f are linearly dependent; (iii) Z is contained in a hyperplane of ( P N ) ∗ ; (iv) X ∗ is contained in a hyperplane of ( P N ) ∗ ; (v) Up to a projective transformation f depends on at most N variables. There are many classical examples of varieties with vanishing Hessianwhich are not cones. The following example appears in the work of Gordanand Noether [GN] and Perazzo [P], called un esempio semplicissimo . Example 2.5.
Let X = V ( f ) ⊂ P be the irreducible hypersurface givenby f = x x + x x x + x x . We can check that X is not a cone, showing for example the linear indepen-dence between the partial derivatives. But since f f = f is an algebraicrelation among them, X has vanishing Hessian. EVELOPABLE CUBICS IN P AND THE LEFSCHETZ LOCUS IN GOR(1 , , ,
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Linearity of general fibers and focal locus.
In this section werecall some useful results about developable varieties. The main basic factis that the general fiber of the Gauss map is a union of finitely many linearspaces, see [Se, p. 95]. Actually, it has been proved by Zak that the closureof a general fiber is irreducible. For instance, see [Za1, Theorem 2.3] or [FP,p. 87].
Theorem 2.6 ([Se, Za1]) . Let X ⊂ P N be an irreducible projective variety.If X is developable then the closure of a general fiber of γ is a linear subspace. Let X ⊂ P N be an irreducible projective variety of dimension n ≥ X is a hypersurface with vanishing Hessian, the fibers of its polarmap share this phenomenon of linearity. We state this result below, theproof can be found in [Za2, Proposition 4.9]. Theorem 2.7 ([Za2]) . Let X ⊂ P N be a reduced hypersurface with vanishingHessian. The closure of the fiber of Φ f over a general point z ∈ Z is a unionof finitely many linear subspaces passing through the subspace ( T z Z ) ∗ . In the next example we illustrate the linearity of fibers of γ and Φ f . Beforemoving on, we recall the definition of a scroll surface S (1 , ⊂ P . Let l ⊂ P be a line and let C ⊂ P be a conic lying on a plane complementary to l .Given an isomorphism ϕ : l −→ C we let S (1 ,
2) be the union of the lines x, ϕ ( x ) joining points of l to corresponding points of C . We note that allthe scrolls S (1 ,
2) are projectively equivalent, see for example [H, Example8.17].
Example 2.8.
We want to describe the fibers of γ and Φ f where f = x x + x x x + x x . In particular, we will see in this example that X ∗ ⊂ ( P ) ∗ isa scroll surface S (1 , X = V ( f ), with reduced structure, is Y = V ( x , x ).Its polar image is the quadratic cone Z = V ( y y − y ) ⊂ ( P ) ∗ which hasas vertex the line l = Y ∗ . Therefore Z ∗ is a conic contained in Y . Considerthe plane P = V ( y , y ) ⊂ ( P ) ∗ , observe that C = Z ∩ P is a conic and Z is the join between C and l .We denote by P t , t ∈ P the family of hyperplanes containing Y and foreach t ∈ P let η t ∈ ( P ) ∗ be the corresponding point of l . The reader cancheck that P t ∩ X is a union of a plane P t and Y , where Y appears withmultiplicity two. A direct calculation shows that for a general point x ∈ P t the closure of Φ − f ( y ), y = Φ f ( x ), is a line contained in P t and passingthrough ξ t = ( T y Z ) ∗ ∈ Z ∗ . In particular, for a general x ∈ P t the closure of γ − ( y ) is a line contained in P t passing through ξ t . Hence X is swept outby planes P t , and fibers of γ lying in P t determine a star of lines passingthrough the point ξ t ∈ Z ∗ .Now we prove that X ∗ is a scroll surface S (1 , x ∈ P t , the tangent space T x X contains P t . Therefore the image of P t by γ is ( P t ) ∗ ∼ = l ′ t . Let µ t ∈ ( P ) ∗ be the point corresponding to the unique FASSARELLA, FERRER, AND GONDIM hyperplane H t containing P t and P ∗ (as P ∗ ⊂ H t , µ t ∈ Z ∩ P = C ). Observethat l ′ t is the line passing through η t ∈ l and through µ t ∈ C . This showsthat X ∗ is a scroll S (1 ,
2) which has as rulings the lines passing through η t ∈ Y ∗ = l and µ t ∈ C , t ∈ P . Figure 1.
Cubic hypersuface with vanishing Hessian.Let X ⊂ P N be a developable projective variety. By Theorem 2.6, X isruled by linear subspaces (fibers of the Gauss map) of dimension k , where k = dim( X ) − dim( X ∨ ) . Let U ⊂ X be the open subset where γ has maximal rank. For each x ∈ U , let L x be the k -dimensional subspace passing through x such that γ isconstant along L x .We will denote by B γ the closure in G ( k, N ) of the set { L x : x ∈ U } .We shall say that B γ is the family of k –dimensional subspaces determinedby fibers of γ . Let B ′ γ be a desingularization of B γ and I ⊂ B ′ γ × P N theincidence variety of B ′ γ with natural projection ψ : I −→ X. For a general x ∈ X the fiber ψ − ( x ) coincides with the point ( L x , x ) ∈ I .Let R ψ be the ramification divisor of ψ and π : I −→ B ′ γ the naturalprojection on the first coordinate. We can write R ψ = H ψ + V ψ wherethe restriction of π to any irreducible component of the support of H ψ isdominant and of the support of V ψ is nondominant. We say that H ψ is the horizontal divisor and V ψ is the vertical divisor . The direct image by ψ ofthe horizontal divisor, denoted by ∆ = ψ ∗ ( H ψ ), is called the focal locus of X .We note that the restriction of ψ to a general fiber of πψ | π − ( L ) : π − ( L ) −→ L is an isomorphism. So the restriction of H ψ to π − ( L ) defines a divisor in L which coincides with the restriction of the focal locus of X to L . Thisdivisor will be denoted by ∆ L .One of the main results concerning developable varieties is the following.For the proof see [IL, Theorem 3.4.2]. EVELOPABLE CUBICS IN P AND THE LEFSCHETZ LOCUS IN GOR(1 , , ,
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Theorem 2.9.
Let X ⊂ P N be a developable projective variety. If X isnot a linear subspace, then X is singular and its focal locus is contained in Sing( X ) . Moreover, for a general L belonging to B γ , the restriction of thefocal locus to L is a divisor ∆ L in L of degree dim( B γ ) . Cubics with vanishing Hessian.
Revisiting the work of Perazzo [P],in [GR] the authors provide a classification of cubic hypersurfaces with van-ishing Hessian in P N , for N ≤
6. In this section we rebuild the classificationfor N = 4. This digression will be useful in the next section. Lemma 2.10.
Let X ⊂ P N , N ≥ , be an irreducible cubic hypersurface.Assume there is a component Y of Sing( X ) , with dim Y = dim X − . Then Y is a linear subspace.Proof. Since X has degree 3, the secant variety S ( Y ) of Y must be containedin X . Hence either S ( Y ) = Y , in this case Y is a linear subspace, or S ( Y ) = X . But the second case cannot occur, because the equalitydim S ( Y ) = dim Y + 1implies that X = S ( Y ) is a linear subspace, see [Ru, Proposition 1.2.2]. (cid:3) Remark 2.11.
Here, we note that an irreducible developable cubic surface X ⊂ P N is a cone. It is well known that a developable surface X ⊂ P N mustbe either a cone or the tangent developable to a curve, see [IL, Theorem3.4.6]. In the last case the curve lies in the singular locus of X . Consequently,it follows from Lemma 2.10 that any developable cubic surface in P is acone. Now, let N ≥
3. By taking the smaller linear subspace containing X , we can assume that X ⊂ P N is non-degenerate, and let H ≃ P N − bea general hyperplane. We will show that N = 3. Since C = X ∩ H is anirreducible non-degenerate curve in H of degree 3, we conclude that N ≤ N = 4, then C is the rational normal curve in H and X is smooth, but since developable (non-linear) varieties are singular,see [Theorem 2.9], this leads us to a contradiction. Lemma 2.12.
Let X ⊂ P be an irreducible cubic hypersurface. Assumethat X is not a cone. If Sing( X ) contains a linearly embedded P then X isprojectively equivalent to V ( f ) , where f = x x + x x x + x x .Proof. Let us suppose W = V ( x , x ) ⊂ Sing( X ). If f is an irreduciblepolynomial defining X one can write f = ax + bx , where a and b arepolynomials of degree two. Since the derivatives of f must vanish in W wecan write f = l x + l x x + l x where l i , i = 0 , ,
2, are linear forms. If X is not a cone, then l , l and l are linearly independent, so there is a projective transformation such that f = x x + x x x + x x . (cid:3) FASSARELLA, FERRER, AND GONDIM
The following proposition is a classical result of Perazzo [P].
Proposition 2.13.
Let X ⊂ P be a cubic hypersurface. Assume that X isnot a cone. The following conditions are equivalent: (i) X has vanishing Hessian;(ii) X ∗ is projectively equivalent to the scroll surface S (1 , Proof.
First we will show that we can assume X irreducible. Suppose that X = V ( f ) is reducible and is not a cone, then f = hq where h is a linearform and q is homogeneous of degree 2. In this case its polar map Φ f isdominant, as follows by a straightforward computation. This also can beproved (when K = C ) by using the following identity d ( X ) = d ( V ( h )) + d ( V ( q )) + d ( V ( h ) ∩ V ( q ))where d ( V ) denotes the degree of the polar map associated to V , see [FM,Corollary 4.3]. Since V ( h ) ∩ V ( q ) is a smooth conic, recall that we areassuming X is not a cone, then the right side of the identity is positive,which implies that Φ f is dominant.Now we suppose that X is irreducible and has vanishing Hessian. ByProposition 2.2, we get Z ∗ ⊂ Sing X . We can assume dim( Z ∗ ) ≥
1, other-wise Z is contained in a hyperplane and Proposition 2.4 ensures that X isa cone.We will show that Z ∗ cannot be a component of Sing( X ). Let us considerthe Perazzo map P X : P N G (codim Z − , N ) x ( T Φ f ( x ) Z ) ∗ . Since X is an irreducible cubic hypersurface, the closure of a general fiber of P X is a linear space, see [GR, Theorem 2.5]. According with [GR, Proposi-tion 2.16] this implies that Z ∗ lies in the intersection of fibers of P X . Andfrom [GR, Proposition 2.13] this is equivalent to say that the linear span < Z ∗ > lies in Sing( X ). Finally, this ensures that if Z ∗ is a component ofSing( X ) then Z ∗ = < Z ∗ > , which implies that Z ∗ is a linear subspace. But,in this case X must be a cone.So far we have proved that dim Z ∗ ≥ Z ∗ cannot be component ofSing X . Hence, one may assume that Sing( X ) contains a two–dimensionalcomponent. It follows from Lemma 2.10 and Lemma 2.12 that X is projec-tively equivalent to V ( f ), where f = x x + x x x + x x . This is enoughto conclude that X ∗ ≃ S (1 , S (1 ,
2) are projectively equiv-alent, X ∗ ≃ S (1 ,
2) implies that X is projectively equivalent to V ( f ), f = x x + x x x + x x . (cid:3) EVELOPABLE CUBICS IN P AND THE LEFSCHETZ LOCUS IN GOR(1 , , ,
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Sections of the secant variety of the Veronese surface.
Linearsections of the secant variety of the Veronese surface are examples of de-velopable cubic hypersurfaces in P ; there are three classes of them, up toprojectivity, which we now describe.Let V be the three-dimensional vector space K . The algebraic groupGL(3) acts on V and consequently induces an action of PGL(3) on P = P (Sym V ). We can also consider the action of PGL(3) on the dual space( P ) ∗ = P (Sym V ∗ ) which we see as the space of conics in P . The image V of the Veronese map v : P ( V ) −→ P (Sym V ) l l is the Veronese surface. Using coordinates ( a : b : c ) of P , the map abovecan be given by ( a : b : c ) ( a : b : c : ab : ac : bc )and if we identify P with the projectivization of the space of symmetricmatrices, by sending each point ( x : · · · : x ) of P to x x x x x x x x x then V coincides with the locus of matrices of rank one. We note thatits secant variety S ( V ) is a cubic hypersurface. Indeed, since any singularmatrix can be written as a sum of two rank one matrices then we see that S ( V ) corresponds to the locus of singular matrices.We now describe V ∗ and the linear sections of V . The preimage by v of ahyperplane in P is a conic in P . The hyperplane is tangent to V if and onlyif the conic is singular, thus the dual variety V ∗ ⊂ ( P ) ∗ of V is isomorphic tothe locus of singular conics in P . Since this last coincides with the locus ofsingular symmetric matrices, we conclude that V ∗ ≃ S ( V ). A pair of distinctlines in P comes from a hyperplane which is tangent to V at a single pointand whose intersection with V yields a pair of conics sharing a single point.A double line in P corresponds to a hyperplane which is tangent along aconic.Since V ∗ ≃ S ( V ) then by biduality we conclude that S ( V ) ∗ ≃ V . Inaddition, the action of PGL(3) on ( P ) ∗ gives three orbits: • U = ( P ) ∗ \V ∗ , yielding sections which are transverse to V ; • U = V ∗ \ S ( V ) ∗ , corresponding to sections which are tangent to V ata single point; and • the closed orbit S ( V ) ∗ , giving sections which are tangent to V alonga conic.Given H ∈ ( P ) ∗ we set X = H ∩ S ( V ) for the corresponding linear sectionof S ( V ). From the discussion above, we obtain the following possibilities: (1) if H ∈ U then H ∩ V is a rational normal curve in P and X is thesecant variety of this curve;(2) if H ∈ U then H ∩ V is a union of two irreducible conics C and C sharing a single point, and X is projectively equivalent to the join S ( C , C ) between them;(3) if H ∈ S ( V ) ∗ then X has vanishing Hessian and it is projectivelyequivalent to V ( f ), where f = x x + x x x + x x .Our next goal is to prove Theorem 1.1, for this purpose we will need somepreliminary lemmas.2.5. Preliminary Lemmas.
Let B γ be the family of linear subspaces de-termined by fibers of the Gauss map γ . If the dual variety X ∗ of X hasdimension two, then B γ is a two–dimensional family of lines. Theorem 2.9ensures that the restriction of the focal locus ∆ of X to a general line L belonging to B γ is a divisor ∆ L of degree two in L . If X ∗ has dimension onethen B γ is a 1-dimensional family of 2-linear subspaces. Applying Theorem2.9 again, we see that ∆ L is a divisor of degree one in L . The next resultswill be useful in the proof of Theorem 1.1. Lemma 2.14.
Let X ⊂ P be an irreducible developable cubic hypersurfacewith nonvanishing Hessian, then X ∗ has dimension .Proof. Since X is not a linear subspace, then the dimension of X ∗ is atleast one. We will show that if X ∗ has dimension 1 then X has vanishingHessian. Assume that dim X ∗ = 1. For a general element L ∈ B γ , ∆ L isa divisor of degree one in L and ∆ is contained is Sing( X ). If ∆ L ≃ P varies with L , then the dimension of the singular set of X is at least two.Lemma 2.10 and Lemma 2.12 will imply that X has vanishing Hessian. If∆ L is a fixed line, say l ≃ P , when L varies in B γ then we will show that X must be a cone whose vertex contains l . This contradicts our hypothesis.Given y ∈ l , let z ∈ S ( y, X ) general: z ∈ < y, x > for general x ∈ X . Weare assuming that the linear subspace L x ∈ B γ passing trough x contains l . In particular < y, x > ⊂ L x ⊂ X . This implies that z ∈ X . Since z ∈ S ( y, X ) is general, we get S ( y, X ) ⊂ X . This shows that S ( y, X ) = X and consequently y ∈ Vert( X ), which is a contradiction. Therefore X ∗ hasdimension 2 and this concludes the proof. (cid:3) Lemma 2.15.
Let X ⊂ P be an irreducible developable cubic hypersurfacesuch that the support of the focal locus ∆ is an irreducible curve. Besides,assume that X ∗ has dimension two. If the restriction of ∆ to a generalline belonging to B γ is one point of multiplicity two, then X has vanishingHessian.Proof. Let us denote by C the support of ∆. Given a general point x ∈ C ,let V x be the cone determined by lines of B γ passing through it. Notice that X is a union of these V x when x varies in C . The developable hypothesison X implies that these cones are tangent planes to C , see [MT, p. 454].Therefore C cannot be a line, otherwise X would be a cone. EVELOPABLE CUBICS IN P AND THE LEFSCHETZ LOCUS IN GOR(1 , , ,
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Since X has degree three, the secant variety of C , S ( C ) is contained in X . Now we analyze the dimension of S ( C ). If it has dimension three, then S ( C ) = X and by Terracini Lemma ∆ L coincides with two distinct points,this contradicts our hypothesis on ∆ L . If the dimension of S ( C ) equals two,we will show that S ( C ) is contained in the singular set of X , which impliesthat X has vanishing Hessian. So, assume thatdim( S ( C )) = 2 = dim( C ) + 1 . In this situation, S ( C ) is a linearly embedded P , see [Ru, Proposition 1.2.2].Suppose that S ( C ) ≃ P is not contained in Sing( X ). Let q ∈ S ( C ) be asmooth point of X and take a tangent line l x of C at x ∈ C passing through q . Since q belongs to the plane V x , the tangent space T q X must contain V x .Thus, T q X is the join between S ( C ) and V x , that is T q X = S ( S ( C ) , V x ) . Hence, the tangent space of X is constant along l x . But, two lines l x and l x ′ , for distinct points x and x ′ , must intersect at one point. Thereforethe tangent space of X is constant along S ( C ). If H denotes the tangentspace at one general point q ∈ S ( C ), then we have V x ⊂ H for a generalpoint x ∈ C . In this case, we must have H = X and this contradicts ourhypothesis deg X = 3.Therefore S ( C ) ≃ P is contained in Sing( X ). Lemma 2.10 and Lemma2.12 imply that X has vanishing Hessian. (cid:3) In order to prove Theorem 1.1 we also need the following result.
Lemma 2.16.
Let C ⊂ P be a non-degenerate irreducible curve whosesecant variety, S ( C ) , is a cubic hypersurface. Then C is a rational normalquartic curve.Proof. It is enough to show that deg C = 4. Let x ∈ C be a smooth point, L = T x C the tangent line at x and P = P ⊂ P a linear space skew to L ,that is, P ∩ L = ∅ . We consider the projection π : C P from L whichsends y ∈ C \ ( C ∩ L ) to π ( y ) = < L, y > ∩ P. Note that ˜ C = S ( L, C ) ∩ P is the closure of the image of C by π . We willshow that ˜ C has degree 2 and this implies that C has degree 4.Let C x be the tangent cone of S ( C ) at x . It has S ( L, C ) as an irreduciblecomponent, see [CR, Theorem 3.1]. Since deg S ( C ) = 3, we get deg C x = 2.But C x cannot be decomposed as product of hyperplanes, otherwise C wouldbe degenerate. This shows that S ( L, C ) = C x and therefore ˜ C has degree 2.This concludes the proof. (cid:3) Proof of Theorem 1.1.
Assume that X ⊂ P is a developable irre-ducible cubic hypersurface which is not a cone. If X has vanishing Hessian,then Proposition 2.13 yields X ∗ ≃ S (1 , X corresponds to asection of S ( V ) which is tangent to V along a conic (see Section 2.4). Suppose that X has nonvanishing Hessian. By Lemma 2.14 we get that X ∗ has dimension two. Thus, B γ is a two–dimensional family of lines. Asconsequence of Theorem 2.9, the restriction of the focal locus to a generalline L ∈ B γ is a divisor of degree two in L .We first remark that any irreducible component of the support | ∆ | of ∆has dimension one. In fact, if there exists a zero dimensional component,say x ∈ X , then X must be a cone because every line L ∈ B γ must passthrough x . Besides that, since ∆ ⊂ Sing( X ) then from Lemma 2.10 andLemma 2.12 the existence of a two–dimensional component of | ∆ | will implythat X has vanishing Hessian.The focal locus ∆ is the direct image of the horizontal divisor H ψ . Recallthat the restriction of ψ : I −→ X to π − ( L ) gives an isomorphism π − ( L ) ≃ L , for general L . The restriction of H ψ to π − ( L ) is a divisor of degree twowhich corresponds to ∆ L , via this isomorphism. Therefore, the supportof H ψ has at most two irreducible components. A fortiori, the number ofirreducible components of | ∆ | is at most two.We will see that if | ∆ | has two irreducible components then it is a linearsection of S ( V ), X = H ∩ S ( V ) with H ∈ V ∗ \ S ( V ) ∗ . Suppose | ∆ | is a unionof two distinct irreducible curves, say C and C . Hence X must be the joinbetween them, X = S ( C , C ). We first remark that C and C are planecurves. Indeed, if for example S ( C ) has dimension 3 then X = S ( C ) and | ∆ | = C , contradicting our hypothesis on ∆. If C and C are disjoint, onehas (see [H, p. 235])3 = deg( X ) = deg( C )deg( C )which means that at least one of these curves is a line and then X is a cone.Let us suppose that C and C are not disjoint and have degree at leasttwo. The two planes containing C and C must share exactly one point.Otherwise, X coincides with the P spanned by them. We denote by p theintersection point of C and C . Now we proceed with the same argumentof [H, p. 236 Calculation II ]. If Γ ⊂ P is a general line, we may describethe intersection Γ ∩ X by considering a general projection π Γ : P P from Γ. Let ˜ C i ⊂ P be the image of C i by π Γ , i = 1 ,
2, and q = π Γ ( p ). Thepoints of Γ ∩ X correspond to the points of ˜ C ∩ ˜ C distinct of q . We notethat the intersection outside q is transverse, thus3 = deg( X ) = deg( C )deg( C ) − I q (1)where I q = I ( ˜ C , ˜ C ; q ) denotes the intersection multiplicity at q . Since C and C do not share a tangent line, for a general choice of Γ we may assumethe same for ˜ C and ˜ C . Then I q is the product of the algebraic multiplicityof ˜ C and ˜ C at q , say I q = a a . By the inequality a i ≤ deg( C i ) − C ) + deg( C ) ≤ . EVELOPABLE CUBICS IN P AND THE LEFSCHETZ LOCUS IN GOR(1 , , ,
1) 13
Hence X is the join between the conics C and C . The reader can check that X is uniquely determined up to a projective transformation. We concludethat X ≃ H ∩ S ( V ) with H ∈ V ∗ \ S ( V ) ∗ and this concludes the case where | ∆ | has two irreducible components.Let us assume that | ∆ | = C is an irreducible curve. If L is a general linebelonging to B γ , we have two possibilities:(1) ∆ L = 2 p ;(2) ∆ L = p + q , with p = q .From Lemma 2.15, the first case cannot happen because we are assumingthat X has nonvanishing Hessian. If we are in case (2), then a general line of B γ is secant to the non-degenerate curve C and then X = S ( C ). By Lemma2.16, X = S ( C ) where C is a rational normal curve. This corresponds tothe case where X ≃ H ∩ S ( V ) where H ∈ ( P ) ∗ \ V ∗ . This finishes the proofof Theorem 1.1.3. The Lefschetz locus in
GOR(1 , , , Artinian Gorenstein algebras and the Lefschetz property.
Let A = d M i =0 A i , be a graded Artinian K -algebra with A d = 0, we say that A is standard graded if A = K and A is generated by A as algebra. Theinteger d is called the socle degree of A . The codimension of A coincideswith its embedding dimension, that is dim A . If A = K [ X , . . . , X N ] /I is astandard graded Artinian K -algebra, where I is an ideal with I = 0, thencodim A = N + 1. The Hilbert vector of A is Hilb( A ) = (1 , a , . . . , a d ), where a k = dim K A k .It is a well known fact that A is a Gorenstein algebra if and only ifdim K A d = 1 and the restrictions of the multiplication in A to comple-mentary degrees A k × A d − k → A d is a perfect pairing, see [HW, Theorem2.79]. For standard graded Artinian Gorenstein algebras, the Hilbert vectoris symmetric: a i = a d − i .We say that a standard graded Artinian Gorenstein algebra A has the Strong Lefschetz Property , or simply A has the SLP, if there exists a linearform L ∈ A such that • L d − i : A i → A d − i is an isomorphism for every 0 ≤ i ≤ ⌊ d ⌋ . In this case L is called a strongLefschetz element .The following is the model for standard graded Artinian Gorenstein alge-bras. Let R = K [ x , . . . , x N ] be the polynomial ring in N + 1 indeterminatesand Q = K [ X , , . . . , X N ] the associated ring of differential operators. Usingthe classical identification X i := ∂∂x i , the ring R has a natural structure of Q − module. In fact, differentiation induces a natural action Q × R → R ,given by ( α, f ) α ( f ). Let f ∈ R d = K [ x , . . . , x N ] d be a homogeneous polynomial of degree deg( f ) = d ≥
1. We define the annihilator ideal byAnn f = { α ∈ Q : α ( f ) = 0 } ⊂ Q. The homogeneous ideal Ann f of Q is also called the Macaulay dual of f .We define A f = Q Ann f . One can verify that A f is a standard graded Artinian Gorenstein K -algebraof socle degree d , see [MW, Section 1.2]. We assume, without loss of gener-ality, that (Ann f ) = 0. This is equivalent to say that the partial derivativesof f are linearly independent, which means that X = V ( f ) is not a cone.By the theory of inverse systems, we get the following characterization ofstandard graded Artinian Gorenstein K -algebras. It is also called Macaulay-Matlis duality. For a more general discussion of Macaulay-Matlis duality see[HW, Section 2.2], [IK, Section 1.1], [BS, Chapter 10] and [E, Chapter 21]. Theorem 3.1. ( Double annihilator theorem of Macaulay)
Let I be an ideal of Q such that Q/I is a standard graded Artinian K -algebraof socle degree d . Then Q/I is Gorenstein if and only if there exists f ∈ R d such that I = Ann f . Let f ∈ R d be a homogeneous polynomial and A f = Q Ann f = d M i =0 A i thestandard graded Artinian Gorenstein algebra associated to f .Let { α , . . . , α s } be an ordered K -basis of A k , with k ≤ d . The k -thHessian of f is the matrixHess kf = [ α i ( α j ( f ))] ≤ i,j ≤ s . Its determinant will be denoted hess kf . Note that Hess f is the classicalHessian Hess f .The following theorem yields a connection between Lefschetz propertiesand higher Hessians. Theorem 3.2. [MW]
Consider A f , where f ∈ R is a homogeneous poly-nomial. An element L = a X + . . . + a N X N ∈ A is a strong Lefschetzelement of A f if and only if hess kf ( a , . . . , a N ) = 0 for all ≤ k ≤ ⌊ d/ ⌋ . Next we discuss the SLP for standard graded Artinian Gorenstein algebrasof socle degree d = 3. For such an algebra the Hilbert vector isHilb( A ) = (1 , N + 1 , N + 1 , . We denote by GOR(1 , N + 1 , N + 1 ,
1) the space of standard graded ArtinianGorenstein algebras of socle degree 3. By Theorem 3.2, A f ∈ GOR(1 , N +1 , N + 1 ,
1) has the SLP if and only if hess f = 0. EVELOPABLE CUBICS IN P AND THE LEFSCHETZ LOCUS IN GOR(1 , , ,
1) 15
By the Gordan-Noether Theorem, if N ≤
3, then hess f = 0 if and only if X = V ( f ) is a cone (see [GN]). Therefore, every standard graded ArtinianGorenstein algebra of socle degree 3 and codimension ≤ A has codimension 5, Proposition 2.13 yields the following result. Proposition 3.3.
Let A be a standard graded Artinian Gorenstein K -algebraof Hilbert vector Hilb( A ) = (1 , , , . Assume that A does not satisfy theSLP. Then A is isomorphic to the following algebra K [ X , X , X , X , X ](( X , X , X ) , X X , X X , X X − X X , X X − X X , ( X , X ) ) . Proof.
The algebra is of the form A f = K [ X , X , X , X , X ] / Ann f , forsome f ∈ K [ x , x , x , x , x ] , not a cone. Theorem 3.2 implies that SLPfails if and only if hess f = 0. By Proposition 2.13, we can assume that f has equation f = x x + x x x + x x . The desired isomorphism can beobtained using this explicit equation. (cid:3) Jordan types.
Let A = d M i =0 A i be a standard graded Artinian K -algebra. For l ∈ A consider the map µ l : A → A given by µ l ( x ) = lx . Since l d +1 = 0, µ l is a nilpotent K -linear map. The Jordan decomposition of sucha map is given by Jordan blocks with 0 in the diagonal, therefore it inducesa partition of dim K A which we denote J A,l . Indeed, the nilpotent linearmap µ l : A → A induces a direct sum decomposition of A into cyclic µ l -invariant subspaces A = m M i =0 C i . The partition J A,l is given by the length k i = dim K C i . Without loss of generality we consider the partition in anon-increasing order.Given a partition P = p ⊕ . . . ⊕ p s of dim K A with p ≥ . . . ≥ p s , wedenote P ∨ the dual partition obtained from P exchanging rows and columnsin the Ferrer diagram (diagram of dots). If P ′ = p ′ ⊕ . . . ⊕ p ′ t is anotherpartition of dim K A with p ′ ≥ . . . ≥ p ′ t , we will write P (cid:22) P ′ and say that P is less than P ′ in the dominance order if for all k we get p + . . . + p k ≤ p ′ + . . . + p ′ k . If the partition P has repeated terms, say f , f , . . . , f r with multiplicity e , e , . . . , e r respectively, we write P = f e ⊕ . . . ⊕ f e r r . Since K is a field of characteristic zero, there is a non empty Zariski opensubset of U ⊂ A where J A,l is constant for l ∈ U , we call it the Jordan type of A and we denote it J A .The following proposition is a special case of [HW, Proposition 3.64]. Itshows that SLP can be described by the Jordan type of A . Proposition 3.4.
Suppose that A = d M i =0 A i is a standard graded Artinian K -algebra with A d = 0 . Then A has the SLP, if and only if, J A = Hilb( A ) ∨ . Since the generic AG algebra in GOR(1 , , ,
1) satisfies the SLP, then thegeneric Jordan type is (1 , , , ∨ = 4 ⊕ . Example 3.5.
The algebra A = Q/ Ann f , f = x x + x x x + x x , hasHilbert vector Hilb( A ) = (1 , , ,
1) and Jordan type J A = 4 ⊕ ⊕ ≺ ⊕ . The following proposition is a consequence of Proposition 3.3 and Exam-ple 3.5.
Proposition 3.6.
Let A be a standard graded Artinian Gorenstein K -algebraof Hilbert vector Hilb( A ) = (1 , N + 1 , N + 1 , . If N ≤ , then A has theSLP. If N = 4 , then the possible Jordan types of A are: either J A = 4 ⊕ if A has the SLP or J A = 4 ⊕ ⊕ if the SLP fails. The Lefschetz locus in
GOR(1 , , , . The affine scheme Gor( T )parametrizing AG algebras with Hilbert vector T was described by A. Iar-robino and V. Kanev and a great account of their work can be found in [IK].In their context, Gor( T ) stands for the affine cone of the projective varietydenoted GOR( T ) here. As we have seen, by Macaulay-Matlis duality thescheme GOR(1 , N + 1 , N + 1 ,
1) can be identified with the parameter spaceof degree 3 homogeneous polynomials f ∈ K [ x , . . . , x N ], up to scalars, suchthat A f has Hilbert vector Hilb( A f ) = (1 , N +1 , N +1 , A f ) = (1 , N + 1 , N + 1 , f ) = 0, i.e. that f in not a cone. Therefore, we havean identification GOR(1 , N + 1 , N + 1 , ≃ P ν ( N ) \ C N where ν ( N ) = (cid:0) N +33 (cid:1) − C N is the parameter space of cubic cones in P N . In particular GOR(1 , , , ≃ P \ C . By Theorem 3.2, an AG algebra A of socle degree 3 has the SLP if andonly if its dual generator f satisfies hess f = 0. Proposition 3.6 gives adescription of their Jordan types. We summarize this discussion in thefollowing proposition. Proposition 3.7.
The space
GOR(1 , N + 1 , N + 1 , can be identified with P ν ( N ) \ C N , where ν = (cid:0) N +33 (cid:1) and C N is the space of cubic cones in P N . For N ≤ all the algebras in GOR(1 , N + 1 , N + 1 , have the SLP. For N = 4 ,the locus in GOR(1 , , , of algebras satisfying SLP is GOR(1 , , , \ Y where Y can be identified with the locus formed by f ∈ P ν ( N ) \ C N withvanishing Hessian. EVELOPABLE CUBICS IN P AND THE LEFSCHETZ LOCUS IN GOR(1 , , ,
1) 17 Parameter spaces
In this section we find parameter spaces for the locus of cubic cones in P , for the locus of cubics with vanishing Hessian and for their intersection;using these descriptions, we compute their dimensions and degrees.To compute the degree our principal tools are the Segre and Chern classesof a vector bundle. We refer the reader to [F, § § V the vector space K . Hence wehave: P = P ( V ), R = Sym V ∗ , P = P ( R ) = P (Sym ( V ∗ )).4.1. Parameter space for cubic cones in P . In this section we find aparameter space for cubic cones in P and compute its dimension and degree.A cubic cone in P is determined by a point x ∈ P and a cubic hypersur-face in the P projectivization of the quotient V /x . Consider the tautologicalsequence on P = P ( V ):(2) 0 → O P ( − → O P ⊗ V → P → . The fiber of P ( P ) over x ∈ P can be identified with P = P ( V /x ). There-fore P (Sym ( P ∗ )) parametrizes cubic hypersurfaces lying in each P , where P ∗ denotes the dual of the vector bundle P . Note that F = Sym ( P ∗ ) is asubbundle of O P ⊗ Sym ( V ∗ ). Thus we have two projections P ( F ) p | | ③③③③③③③③③ p ( ( PPPPPPPPPPPP P P = P (Sym ( V ∗ ))where p is generically injective and C is the image of p . We have thefollowing result. Proposition 4.1.
Let C ⊂ P be the space of cubic cones in P . Thenthe dimension of C is ; its degree is given by the Segre class s ( F ) and isequal to 1365.Proof. The dimension can be computed bydim( C ) = dim( P ( F )) = 4 + rk ( F ) − rk ( F ) = 20.To compute the degree, write H for the hyperplane class of P . We have p ∗ H = c O F (1) =: h . We may computedeg C = Z P H ∩ C = Z P ( F ) h = Z P p ∗ ( h ) = Z P s ( F ) . Using Macaulay2 [GS] we find s ( F ) ∩ P = 1365 (see the Scripts in § (cid:3) This calculation can be generalized to determine the degree of the locus C d,n of cones in the space parametrizing hypersurfaces of degree d in P n .Using another characterization for cones, this degree is calculated in [EH, Proposition 7.8, p. 257]. There the authors obtain the following formula forthe degree of the locus in P ( n + dn ) of cones of degree d :deg( C d,n ) = (cid:18)(cid:0) n + d − n (cid:1) n (cid:19) . Parameter space for cubics with vanishing Hessian.
This sectionis devoted to the description of the parameter space for cubic hypersurfaceswith vanishing Hessian in P = P ( V ).Let H ⊂ P be the locus of hypersurfaces with vanishing Hessian. As inProposition 3.7 we denote Y = H \ C .On the other hand, let K ⊂ P be the locus formed by f ∈ P such that f ∈ I W , where I W is the ideal of some 2-plane W ⊂ P . It is an irreduciblesubvariety of P , actually we will show below that it is the image by amorphism of a projective bundle over G (2 , Remark 4.2.
We claim that Y = K . Note that K ⊂ Y by Lemma 2.12.Reciprocally Proposition 2.13 yields Y ⊂ K . This concludes the claim.We shall find a parameter space for Y using the above characterization.Let G = G (2 ,
4) denote the Grassmannian of 2-planes in P . We have thefollowing tautological sequence on G :(3) 0 → T → O G ⊗ V → Q → T is a subbundle of rank 3 and Q is a bundle of rank 2.Consider the multiplication map ϕ : Sym Q ∗ ⊗ V ∗ → O G ⊗ Sym ( V ∗ ) . It defines a map of vector bundles whose image parametrizes the set of pairs(
W, f ) ∈ G × Sym ( V ∗ ) such that f ∈ I W .We claim that the kernel of ϕ is exactly ∧ Q ∗ ⊗ Q ∗ . Therefore we havean exact sequence(4) 0 → ∧ Q ∗ ⊗ Q ∗ → Sym Q ∗ ⊗ V ∗ ϕ −→ E → E = Im ϕ .Let us prove the claim. Consider the following exact diagram(5) ∧ Q ∗ ⊗ Q ∗ / / / / (cid:15) (cid:15) (cid:15) (cid:15) Kerϕ / / / / (cid:15) (cid:15) (cid:15) (cid:15) Ker ¯ ϕ (cid:15) (cid:15) (cid:15) (cid:15) Sym Q ∗ ⊗ Q ∗ / / / / m (cid:15) (cid:15) (cid:15) (cid:15) Sym Q ∗ ⊗ V ∗ / / / / ϕ (cid:15) (cid:15) (cid:15) (cid:15) Sym Q ∗ ⊗ T ∗ ¯ ϕ (cid:15) (cid:15) (cid:15) (cid:15) Sym Q ∗ / / / / Imϕ / / / / Im ¯ ϕ, EVELOPABLE CUBICS IN P AND THE LEFSCHETZ LOCUS IN GOR(1 , , ,
1) 19
To prove the claim it suffices to prove that ¯ ϕ is injective. Let W ∈ G be a two-plane, as G is a homogeneous variety, we can assume that W = V ( x , x ). Assume ¯ f lies in the fiber of Sym Q ∗ ⊗ T ∗ over W , then it canbe written as ¯ f = x ⊗ f + x x ⊗ f + x ⊗ f where f i are homogeneous polynomials of degree one, f i ∈ h x , x , x i .Hence ¯ ϕ ( ¯ f ) = 0 means that ϕ ( f ) lies in the fiber of Sym Q ∗ , i.e. x f + x x f + x f ∈ Sym ( x , x ), therefore f = f = f = 0.We note that E corresponds to a vector bundle over G whose projectiviza-tion coincides with the incidence variety P ( E ) = { ( W, f ) ∈ G × P : f ∈ I W } . Let us denote by p : P ( E ) −→ G and p : P ( E ) −→ P the natural pro-jections. We see that K is the image of p and p is generically injective.Hence one obtains the following result. Proposition 4.3.
The locus K is the birational image of a projective bundleover the Grassmannian G . The dimension of K is and its degree is ,the degree of the Segre class s ( E ) .Proof. The dimension can be computed asdim( K ) = dim( P ( E )) = dim G + rk ( E ) − . Since rk ( E ) = 13 and dim G = 6 the result follows.We shall write H for the hyperplane class of P . We have p ∗ H = c O E (1) =: h . Using [F, § K is given by Z P ( E ) h = Z G p ∗ ( h ) = Z G s ( E ) . By sequence (4), using the properties of Chern classes [F, § s ( E ) = s (Sym Q ∗ ⊗ V ∗ ) c ( ∧ Q ∗ ⊗ Q ∗ ) = s (Sym Q ∗ ) c ( ∧ Q ∗ ⊗ Q ∗ ) . We can compute these characteristic classes using Macaulay2: s ( E ) = 29960(see the Scripts in § (cid:3) The locus
K ∩ C . In this section we describe the intersection
K ∩ C .Let us consider again the tautological sequence on G = G (2 , → T → O G ⊗ V → Q → . Note that T corresponds to a vector bundle whose projectivization coincideswith the incidence variety P ( T ) = { ( W, p ) ∈ G × P : p ∈ W } . Let q : G × P −→ G and q : G × P −→ P be the natural projections anddenote by π = q | P ( T ) and π = q | P ( T ) their restrictions to P ( T ). We will construct a vector bundle E over P ( T ) with a birational morphism P ( E ) −→ K ∩ C . Consider the tautological sequence on P = P ( V ):0 → O P ( − → O P ⊗ V → P → . Over P ( T ) we have the following multiplication map: ξ : π ∗ Sym Q ∗ ⊗ π ∗ P ∗ → O P ( T ) ⊗ Sym ( V ∗ ) . Its kernel is exactly π ∗ ( ∧ Q ∗ ⊗ Q ∗ ). This can be proved using the sameargument we applied to the multiplication map ϕ that appears in Section4.2. Therefore we have an exact sequence(6) 0 → π ∗ ( ∧ Q ∗ ⊗ Q ∗ ) → Sym π ∗ Q ∗ ⊗ π ∗ P ∗ → E → E = Im ξ . Note that E defines a vector bundle whose fiber over apoint ( W, p ) ∈ P ( T ) is equal to the vector space { f ∈ Sym ( V ∗ ) : f is a cone with vertex p and f ∈ I W } . Let us consider the natural projections p : P ( E ) −→ P ( T ) and p : P ( E ) −→ P . We see that p is generically injective and its image is K ∩ C . Hence one obtains the following result. Proposition 4.4. (1)
K ∩ C is a divisor in K . (2) The degree of
K ∩ C is . It is determined by s ( E ) + ( c (Sym Q ∗ ) + c ( Q )) s ( E ) ∩ [ G ] where E is the vector bundle of Section 4.2.Proof. The dimension of
K ∩ C coincides withdim( P ( E )) = dim P ( T ) + rk ( E ) − . Since rk ( E ) = 10 and dim P ( T ) = 8 the result follows. In order to com-pute the degree, write H for the hyperplane class of P . We have p ∗ H = c O E (1) =: h . The degree of K ∩ C is given by Z P ( E ) h = Z P ( T ) p ∗ ( h ) = Z P ( T ) s ( E ) . We claim that the following identity occurs in the Chow ring of G × P :[ P ( T )] = c ( q ∗ O P (1) ⊗ q ∗ Q ) ∩ [ G × P ] . (7)Indeed, exact sequences (2) and (3) yield a map θ : q ∗ O P ( − θ % % ❑❑❑❑❑❑❑❑❑❑ (cid:15) (cid:15) q ∗ T / / O G × P ⊗ V / / q ∗ Q EVELOPABLE CUBICS IN P AND THE LEFSCHETZ LOCUS IN GOR(1 , , ,
1) 21 which has P ( T ) as zeros. Then it induces a regular section σ : G × P → q ∗ O P (1) ⊗ q ∗ Q which has P ( T ) as zeros. This proves identity (7) (see [F,ex 3.2.16, p. 61]).Putting together sequences (4) and (6) we get s ( E ) = s ( π ∗ E ) c ( G ) . Where G = Sym π ∗ Q ∗ ⊗ π ∗ O P (1). Using this, we deduce: s ( E ) = s ( π ∗ E ) + s ( π ∗ E ) c ( G ) + s ( π ∗ E ) c ( G ) + s ( π ∗ E ) c ( G ) . We note that s i ( π ∗ E ) = 0 for i > E is a vector bundle over G which has dimension 6. From this and (7) one obtains that s ( E ) ∩ [ P ( T )]coincides with( s ( π ∗ E ) c ( G ) + s ( π ∗ E ) c ( G )) ∩ ( k + kc ( π ∗ Q ) + c ( π ∗ Q )) ∩ [ G × P ]where k = c ( q ∗ O P (1)).In what follows we will omit the pull-back. Computing the Chern classesof a tensor product, we obtain (cid:26) c ( G ) = c (Sym Q ∗ ) + 2 c (Sym Q ∗ ) k + 3 k c ( G ) = c (Sym Q ∗ ) + c (Sym Q ∗ ) k + c (Sym Q ∗ ) k + k . (8)Observe that c i ( Q ) s j ( E ) = c i (Sym Q ∗ ) s j ( E ) = 0 , i + j > . Using this and (8), we can reduce the computation of s ( E ) ∩ [ P ( T )] to:( s ( E )3 k + s ( E )( c (Sym Q ∗ ) k + k )) ∩ k ( k + c ( Q )) ∩ [ G × P ] . Finally, since k = 0 and G has dimension 6 we have s ( E ) ∩ [ P ( T )] = 3 s ( E )+( c (Sym Q ∗ ) + c ( Q )) s ( E ) ∩ [ G ] . This number can be computed using Macaulay2/Schubert2, we find 116420.This concludes the proof of proposition. (cid:3)
Now the Theorem 1.2 of the introduction is a consequence of Propositions3.7, 4.1, 4.3 and 4.4.
Scripts. loadPackage "Schubert2"G=flagBundle({1,4})-- Grassmannian of lines in 5-space(S,Q)=G.Bundles-- names the sub and quotient bundles on GR=dual (Q)F=symmetricPower(3,R)--Computes the classes in Proposition 4.1:integral(segre(4,F)) loadPackage "Schubert2"G=flagBundle({3,2})-- Grassmannian of 3-planes in 5-space(S,Q)=G.BundlesR=dual (Q)A=symmetricPower(2,R)B=A^5C=exteriorPower(2,R)*RE=B-C--Computes the classes in Proposition 4.3:integral(segre(6,E))--Computes the classes in Proposition 4.4:integral(3*segre(6,E)+(chern(1,A)+chern(1,Q))*segre(5,E))
Acknowledgments . We wish to thank F. Russo, I. Vainsencher, G.Staglian`o and N. Medeiros for pointing out some mistakes in an early draftand for inspiring conversations on the subject.
References [AG] M. A. Akivis and V. V. Goldberg.
Differential geometry of varieties with degen-erate Gauss maps.
CMS books in Mathematics (2004).[BS] M. P. Brodmann and R. Y. Sharp.
Local cohomology: an algebraic introductionwith geometric applications.
Vol. 136. Cambridge university press (2012).[CG] B. Costa and R. Gondim.
The Jordan type of graded Artinian Gorenstein algebras.
Advances in Applied Mathematics, 111 (2019).[CR] C. Ciliberto and F. Russo,
Varieties with minimal secant degree and linear systemsof maximal dimension on surfaces.
Advances in Mathematics, vol. 200, Number1 (2006), 1–50.[E] D. Eisenbud.
Commutative algebra: with a view toward algebraic geometry.
Vol.150, Springer Science and Business Media (2013).[EH] D. Eisenbud and J. Harris. , Cambridge University Press (2016).[FM] T. Fassarella and N. Medeiros.
On the polar degree of projective hypersurfaces.
J.London Math. Soc. (2012) 86 (1): 259–271.[FP] G. Fischer and J. Piontkowski.
Ruled varieties, an introduction to algebraic dif-ferential geometry.
Advanced Lectures in Mathematics. Friedr. Vieweg & Sohn,Braunschweig (2001).[F] W. Fulton.
Intersection Theory . Springer-Verlag. New York. 1985.[GN] P. Gordan and M. Noether.
Ueber die algebraischen Formen, deren Hesse’schedeterminante identisch verschwindet.
Math. Ann. 10 (1876), 547–568.[GR] R. Gondim and F. Russo.
On cubic hypersurfaces with vanishing hessian.
Journalof Pure and Applied Algebra. 219 (2015), 779–806.[GS] D. Grayson and M. E. Stillman.
Macaulay2 version 1.9.2, asoftware system for research in algebraic geometry.
Available athttps://faculty.math.illinois.edu/Macaulay2/.[GH] P. Griffiths and J. Harris.
Principles of algebraic geometry.
Wiley, New York(1978).
EVELOPABLE CUBICS IN P AND THE LEFSCHETZ LOCUS IN GOR(1 , , ,
1) 23 [HW] T. Harima, T. Maeno, H. Morita, Y. Numata, A. Wachi and J. Watanabe.
TheLefschetz properties.
Lecture Notes in Mathematics 2080. Springer, Heidelberg,2013, xx+250 pp.[H] J. Harris.
Algebraic geometry, a first course.
Springer–Verlag (1992).[IK] A. Iarrobino and V. Kanev.
Power sums, Gorenstein algebras, and determinantalloci . Springer Science and Business Media (1999).[IL] T.A. Ivey and J.M. Landsberg.
Cartan for beginners: differential geometry viamoving frames and exterior differential systems.
Graduate Studies in Mathemat-ics, 61, American Mathematical Society, Providence, RI, xiv+378, (2003).[MW] T. Maeno and J. Watanabe.
Lefschetz elements of Artinian Gorenstein algebrasand Hessians of homogeneous polynomials.
Illinois J. Math. 53 (2009), 593–603.[MT] E. Mezzetti and O. Tommasi.
On projective varieties of dimension n+k coveredby k-spaces.
Illinois J.Math., 46 n º
2, 443–465, (2002).[P] U. Perazzo.
Sulle variet´a cubiche la cui hessiana svanisce identicamente.
Giornaledi Matematiche (Battaglini), (1900) 38, 337–354.[Ru] F. Russo.
On the Geometry of some special projective varieties.
Lecture Notes ofthe Unione Matematica Italiana, vol. 18 Springer (2016).[Se] C. Segre.
Preliminari di una teoria delle varieta luoghi di spazi.
Rend. Curc. Mat.Palermo (1) 30 (1910), 87-121 (JFM 41, p. 724.)[St] R. Stanley.
Weyl groups, the hard Lefschetz theorem, and the Sperner property ,SIAM J. Algebraic Discrete Methods 1 (1980), 168–184.[W] J. Watanabe.
The Dilworth number of Artinian rings and finite posets with rankfunction.
Commutative algebra and combinatorics (Kyoto, 1985), 303?312, Adv.Stud. Pure Math., 11, North-Holland, Amsterdam, 1987.[Za1] F. L. Zak.
Tangents and secants of algebraic varieties.
Transl. Math. Monogr. vol.127, Amer. Math. Soc., U.S.A. (1993).[Za2] F. L. Zak.
Determinants of projective varieties and their degrees.
Encyclopaediaof Mathematical Sciences , vol. 132, Springer Verlag (2004).
Thiago Fassarella, Universidade Federal Fluminense, Rua Alexandre Moura8 - S˜ao Domingos, 24210-200 Niter´oi, Rio de Janeiro, Brasil
Email address : [email protected] Viviana Ferrer, Universidade Federal Fluminense, Rua Alexandre Moura8 - S˜ao Domingos, 24210-200 Niter´oi, Rio de Janeiro, Brasil
Email address : [email protected] Rodrigo Gondim, Universidade Federal Rural de Pernambuco, av. DomManoel de Medeiros s/n, Dois Irm˜aos, 52171-900 Recife, Pernambuco, Brasil
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