Let X be a non-singular quasi-projective variety over a field, and let E be a vector bundle over ... more Let X be a non-singular quasi-projective variety over a field, and let E be a vector bundle over X. Let G X (d, E) be the Grassmann bundle of E over X parametrizing corank d subbundles of E with projection π : G X (d, E) → X, let Q ← π * E be the universal quotient bundle of rank d, and denote by θ the Plücker class of G X (d, E), that is, the first Chern class of the Plücker line bundle, det Q. In this short note, a closed formula for the push-forward of powers of the Plücker class θ is given in terms of the Schur polynomials in Segre classes of E, which yields a degree formula for G X (d, E) with respect to θ when X is projective and ∧ d E is very ample.
The image of the higher Gauss map for a projective variety is discussed. The notion of higher Gau... more The image of the higher Gauss map for a projective variety is discussed. The notion of higher Gauss maps here was introduced by Fyodor L. Zak as a generalization of both ordinary Gauss maps and conormal maps. The main result is a closed formula for the degree of those images of Veronese varieties. This yields a generalization of a classical formula by George Boole on the degree of the dual varieties of Veronese varieties in 1844. As an application of our formula, degree bounds for higher Gauss map images of Veronese varieties are given.
Let X be a non-singular quasi-projective variety over a field, and let E be a vector bundle over ... more Let X be a non-singular quasi-projective variety over a field, and let E be a vector bundle over X. Let G X (d, E) be the Grassmann bundle of E over X parametrizing corank d subbundles of E with projection π : G X (d, E) → X, and let Q ← π * E be the universal quotient bundle of rank d. In this article, a closed formula for π * ch(det Q), the push-forward of the Chern character of the Plücker line bundle det Q by π is given in terms of the Segre classes of E. Our formula yields a degree formula for G X (d, E) with respect to det Q when X is projective and ∧ d E is very ample. To prove the formula above, a push-forward formula in the Chow rings from a partial flag bundle of E to X is given.
O. INTReDgCTION H. Freudenthal constructed, in a series of his papers (see [10] and its reference... more O. INTReDgCTION H. Freudenthal constructed, in a series of his papers (see [10] and its references), the exceptional Lie algebras of type Es, E7, E6 and E4, with defining various projective varieties. The purpose of our work is to study projective geometry for his varieties of certaixx type, which are caJled varieties of planes in the symplectic geometry of Freudenthal (see [10, 4.ll], [24, 2.31). Let g be a graded, simple, finite-dimensional Lie algebra over the complex number field Åë with grades between -2 and 2, dimg2 =-L 1 and gi:O, namely a graded Lie algebra of contact type: g mm gpt2Og-1 ego egi eg2 (see gl). We set Y:--{x Eg X {g}Kad x)2g-2 rm e}, attd defiite an algebraic set V in ge(gD to be the projectivizatioxx of Y: where 7r : gA {O} -P(gi) is the natural projection. Then we cal1 V g P(gi) (with the reduced struÅëture) the FVeudenthal varietgy associated to the graded Lie algebra g of contact type, which is a natural generalization of Freudenthal's varieties mentioned above: Note that V is not necessarily connected in this general setting. We here consider moreover the projectivization of a elosed set {x G g"(ad x)fo+ig-2 == e}, and denote it by Vk; we have e = Ye g va E V2 ! V3 ! Y4 == ge, where we set P := ew(gi) for shert. Clearly, V3 is a quartic hypersurface, V2 is an intersection of cubics and Vi = V is an intersection of quadrics, with a few exceptions. In the literature, several results have been known about the structure of gi as a go-space, case-by-case for each exceptional Lie algebra of types Es, E7, jEi)6 and jF14, from the view-point of the invariant theory of prehomogeneous vector spaces (see [13], [15], [201, [23]). By virtue of those results, it can be shown, for example, that the stratification of P given by the differences of Vk-'s exactly correspends to the orbit decomposition of the go-space gi for those exceptional Me algebras, and also that Freudenthal varieties V asseciated to the algebras of type Es, E7, E6 and F4 gre respectively projectively equlva}eRt to the 27-dimelt$ionai g7-varlety ayisi}ig from the 56--elmeRsic}ickl iyyedgcible yepresent&tioR, t}3e o:gkegeital Gi'assmakit variety gf isotropic 6-planes IR ÅqC22 (Rame}y, tlie 15-dimeit$ion&l spiRor varlety), tke (];ras$maa}m variety ef 3-plaiies iR Åë6 and the symp}ectic Grassmann variety of lsotrepic 3-plalie$ in Åë6, with dimP == 55,31, 19 aftd 13, respectively (see Appendix 1): for those homogeneous projective varieties, we refer to [12, S23.3]. In thSs article we study the IFlreudenthal varieties V with the filtration {Vk} of the ambient space P, from the view-point of projective geometry, not individually but systematically in terms tktwueIaj":U]:tuca y' tziitO •? VA (2oo4110/2't', 9:3010:30am
Abstract. Let X be a non-singular quasi-projective variety over a field, and let E be a vector bu... more Abstract. Let X be a non-singular quasi-projective variety over a field, and let E be a vector bundle over X . Let GX(d, E) be the Grassmann bundle of E overX parametrizing corank d subbundles of E with projection π : GX(d, E) → X , and let Q ← π ∗E be the universal quotient bundle of rank d. In this article, a closed formula for π∗ ch(detQ), the push-forward of the Chern character of the Plücker line bundle detQ by π is given in terms of the Segre classes of E . Our formula yields a degree formula for GX(d, E) with respect to detQ when X is projective and ∧E is very ample. To prove the formula above, a push-forward formula in the Chow rings from a partial flag bundle of E to X is given.
Transactions of the American Mathematical Society, 1999
The secant variety of a projective variety X in P, denoted by Sec X, is defined to be the closure... more The secant variety of a projective variety X in P, denoted by Sec X, is defined to be the closure of the union of lines in P passing through at least two points of X, and the secant deficiency of X is defined by δ := 2 dim X + 1dim Sec X. We list the homogeneous projective varieties X with δ > 0 under the assumption that X arise from irreducible representations of complex simple algebraic groups. It turns out that there is no homogeneous, non-degenerate, projective variety X with Sec X = P and δ > 8, and the E 6 -variety is the only homogeneous projective variety with largest secant deficiency δ = 8. This gives a negative answer to a problem posed by R. Lazarsfeld and A. Van de Ven if we restrict ourselves to homogeneous projective varieties.
Let X be a non-singular quasi-projective variety over a field, and let E be a vector bundle over ... more Let X be a non-singular quasi-projective variety over a field, and let E be a vector bundle over X. Let G X (d, E) be the Grassmann bundle of E over X parametrizing corank d subbundles of E with projection π : G X (d, E) → X, let Q ← π * E be the universal quotient bundle of rank d, and denote by θ the Plücker class of G X (d, E), that is, the first Chern class of the Plücker line bundle, det Q. In this short note, a closed formula for the push-forward of powers of the Plücker class θ is given in terms of the Schur polynomials in Segre classes of E, which yields a degree formula for G X (d, E) with respect to θ when X is projective and ∧ d E is very ample.
An adjoint variety X(g)associated to a complex simple Lie algebra is by definition a projective v... more An adjoint variety X(g)associated to a complex simple Lie algebra is by definition a projective variety in ℙ*(g) obtained as the projectivization of the (unique) non-zero, minimal nilpotent orbit in g. We first describe the tangent loci of X(g) in terms of triples. Secondly for a graded decomposition of contact type we show that the intersection of X(g) and the linear subspace ℙ*(g1) in ℙ*(g) coincides with the cubic Veronese variety associated to g.
The image of the higher Gauss map for a projective variety is discussed. The notion of higher Gau... more The image of the higher Gauss map for a projective variety is discussed. The notion of higher Gauss maps here was introduced by Fyodor L. Zak as a generalization of both ordinary Gauss maps and conormal maps. The main result is a closed formula for the degree of those images of Veronese varieties. This yields a generalization of a classical formula by George Boole on the degree of the dual varieties of Veronese varieties in 1844. As an application of our formula, degree bounds for higher Gauss map images of Veronese varieties are given.
The reflexivity, the (semi-)ordinariness, the dimension of dual varieties and the structure of Ga... more The reflexivity, the (semi-)ordinariness, the dimension of dual varieties and the structure of Gauss maps are discussed for Segre varieties, where a Segre variety is the image of the product of two or more projective spaces under Segre embedding. A generalization is given to a theorem of A. Hefez and A. Thorup on Segre varieties of two projective spaces. In particular, a new proof is given to a theorem of F. Knop, G. Menzel, I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky that states a necessary and sufficient condition for Segre varieties to have codimension one duals. On the other hand, a negative answer is given to a problem raised by S. Kleiman and R. Piene as follows: For a projective variety of dimension at least two, do the Gauss map and the natural projection from the conormal variety to the dual variety have the same inseparable degree?
Let X be a non-singular quasi-projective variety over a field, and let E be a vector bundle over ... more Let X be a non-singular quasi-projective variety over a field, and let E be a vector bundle over X. Let G X (d, E) be the Grassmann bundle of E over X parametrizing corank d subbundles of E with projection π : G X (d, E) → X, let Q ← π * E be the universal quotient bundle of rank d, and denote by θ the Plücker class of G X (d, E), that is, the first Chern class of the Plücker line bundle, det Q. In this short note, a closed formula for the push-forward of powers of the Plücker class θ is given in terms of the Schur polynomials in Segre classes of E, which yields a degree formula for G X (d, E) with respect to θ when X is projective and ∧ d E is very ample.
The image of the higher Gauss map for a projective variety is discussed. The notion of higher Gau... more The image of the higher Gauss map for a projective variety is discussed. The notion of higher Gauss maps here was introduced by Fyodor L. Zak as a generalization of both ordinary Gauss maps and conormal maps. The main result is a closed formula for the degree of those images of Veronese varieties. This yields a generalization of a classical formula by George Boole on the degree of the dual varieties of Veronese varieties in 1844. As an application of our formula, degree bounds for higher Gauss map images of Veronese varieties are given.
Let X be a non-singular quasi-projective variety over a field, and let E be a vector bundle over ... more Let X be a non-singular quasi-projective variety over a field, and let E be a vector bundle over X. Let G X (d, E) be the Grassmann bundle of E over X parametrizing corank d subbundles of E with projection π : G X (d, E) → X, and let Q ← π * E be the universal quotient bundle of rank d. In this article, a closed formula for π * ch(det Q), the push-forward of the Chern character of the Plücker line bundle det Q by π is given in terms of the Segre classes of E. Our formula yields a degree formula for G X (d, E) with respect to det Q when X is projective and ∧ d E is very ample. To prove the formula above, a push-forward formula in the Chow rings from a partial flag bundle of E to X is given.
O. INTReDgCTION H. Freudenthal constructed, in a series of his papers (see [10] and its reference... more O. INTReDgCTION H. Freudenthal constructed, in a series of his papers (see [10] and its references), the exceptional Lie algebras of type Es, E7, E6 and E4, with defining various projective varieties. The purpose of our work is to study projective geometry for his varieties of certaixx type, which are caJled varieties of planes in the symplectic geometry of Freudenthal (see [10, 4.ll], [24, 2.31). Let g be a graded, simple, finite-dimensional Lie algebra over the complex number field Åë with grades between -2 and 2, dimg2 =-L 1 and gi:O, namely a graded Lie algebra of contact type: g mm gpt2Og-1 ego egi eg2 (see gl). We set Y:--{x Eg X {g}Kad x)2g-2 rm e}, attd defiite an algebraic set V in ge(gD to be the projectivizatioxx of Y: where 7r : gA {O} -P(gi) is the natural projection. Then we cal1 V g P(gi) (with the reduced struÅëture) the FVeudenthal varietgy associated to the graded Lie algebra g of contact type, which is a natural generalization of Freudenthal's varieties mentioned above: Note that V is not necessarily connected in this general setting. We here consider moreover the projectivization of a elosed set {x G g"(ad x)fo+ig-2 == e}, and denote it by Vk; we have e = Ye g va E V2 ! V3 ! Y4 == ge, where we set P := ew(gi) for shert. Clearly, V3 is a quartic hypersurface, V2 is an intersection of cubics and Vi = V is an intersection of quadrics, with a few exceptions. In the literature, several results have been known about the structure of gi as a go-space, case-by-case for each exceptional Lie algebra of types Es, E7, jEi)6 and jF14, from the view-point of the invariant theory of prehomogeneous vector spaces (see [13], [15], [201, [23]). By virtue of those results, it can be shown, for example, that the stratification of P given by the differences of Vk-'s exactly correspends to the orbit decomposition of the go-space gi for those exceptional Me algebras, and also that Freudenthal varieties V asseciated to the algebras of type Es, E7, E6 and F4 gre respectively projectively equlva}eRt to the 27-dimelt$ionai g7-varlety ayisi}ig from the 56--elmeRsic}ickl iyyedgcible yepresent&tioR, t}3e o:gkegeital Gi'assmakit variety gf isotropic 6-planes IR ÅqC22 (Rame}y, tlie 15-dimeit$ion&l spiRor varlety), tke (];ras$maa}m variety ef 3-plaiies iR Åë6 and the symp}ectic Grassmann variety of lsotrepic 3-plalie$ in Åë6, with dimP == 55,31, 19 aftd 13, respectively (see Appendix 1): for those homogeneous projective varieties, we refer to [12, S23.3]. In thSs article we study the IFlreudenthal varieties V with the filtration {Vk} of the ambient space P, from the view-point of projective geometry, not individually but systematically in terms tktwueIaj":U]:tuca y' tziitO •? VA (2oo4110/2't', 9:3010:30am
Abstract. Let X be a non-singular quasi-projective variety over a field, and let E be a vector bu... more Abstract. Let X be a non-singular quasi-projective variety over a field, and let E be a vector bundle over X . Let GX(d, E) be the Grassmann bundle of E overX parametrizing corank d subbundles of E with projection π : GX(d, E) → X , and let Q ← π ∗E be the universal quotient bundle of rank d. In this article, a closed formula for π∗ ch(detQ), the push-forward of the Chern character of the Plücker line bundle detQ by π is given in terms of the Segre classes of E . Our formula yields a degree formula for GX(d, E) with respect to detQ when X is projective and ∧E is very ample. To prove the formula above, a push-forward formula in the Chow rings from a partial flag bundle of E to X is given.
Transactions of the American Mathematical Society, 1999
The secant variety of a projective variety X in P, denoted by Sec X, is defined to be the closure... more The secant variety of a projective variety X in P, denoted by Sec X, is defined to be the closure of the union of lines in P passing through at least two points of X, and the secant deficiency of X is defined by δ := 2 dim X + 1dim Sec X. We list the homogeneous projective varieties X with δ > 0 under the assumption that X arise from irreducible representations of complex simple algebraic groups. It turns out that there is no homogeneous, non-degenerate, projective variety X with Sec X = P and δ > 8, and the E 6 -variety is the only homogeneous projective variety with largest secant deficiency δ = 8. This gives a negative answer to a problem posed by R. Lazarsfeld and A. Van de Ven if we restrict ourselves to homogeneous projective varieties.
Let X be a non-singular quasi-projective variety over a field, and let E be a vector bundle over ... more Let X be a non-singular quasi-projective variety over a field, and let E be a vector bundle over X. Let G X (d, E) be the Grassmann bundle of E over X parametrizing corank d subbundles of E with projection π : G X (d, E) → X, let Q ← π * E be the universal quotient bundle of rank d, and denote by θ the Plücker class of G X (d, E), that is, the first Chern class of the Plücker line bundle, det Q. In this short note, a closed formula for the push-forward of powers of the Plücker class θ is given in terms of the Schur polynomials in Segre classes of E, which yields a degree formula for G X (d, E) with respect to θ when X is projective and ∧ d E is very ample.
An adjoint variety X(g)associated to a complex simple Lie algebra is by definition a projective v... more An adjoint variety X(g)associated to a complex simple Lie algebra is by definition a projective variety in ℙ*(g) obtained as the projectivization of the (unique) non-zero, minimal nilpotent orbit in g. We first describe the tangent loci of X(g) in terms of triples. Secondly for a graded decomposition of contact type we show that the intersection of X(g) and the linear subspace ℙ*(g1) in ℙ*(g) coincides with the cubic Veronese variety associated to g.
The image of the higher Gauss map for a projective variety is discussed. The notion of higher Gau... more The image of the higher Gauss map for a projective variety is discussed. The notion of higher Gauss maps here was introduced by Fyodor L. Zak as a generalization of both ordinary Gauss maps and conormal maps. The main result is a closed formula for the degree of those images of Veronese varieties. This yields a generalization of a classical formula by George Boole on the degree of the dual varieties of Veronese varieties in 1844. As an application of our formula, degree bounds for higher Gauss map images of Veronese varieties are given.
The reflexivity, the (semi-)ordinariness, the dimension of dual varieties and the structure of Ga... more The reflexivity, the (semi-)ordinariness, the dimension of dual varieties and the structure of Gauss maps are discussed for Segre varieties, where a Segre variety is the image of the product of two or more projective spaces under Segre embedding. A generalization is given to a theorem of A. Hefez and A. Thorup on Segre varieties of two projective spaces. In particular, a new proof is given to a theorem of F. Knop, G. Menzel, I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky that states a necessary and sufficient condition for Segre varieties to have codimension one duals. On the other hand, a negative answer is given to a problem raised by S. Kleiman and R. Piene as follows: For a projective variety of dimension at least two, do the Gauss map and the natural projection from the conormal variety to the dual variety have the same inseparable degree?
Uploads
Papers by hajime kaji