Abstract.We determine the varieties of linear spaces on rational homogeneous varieties, provide explicit geometric models for these spaces, and establish basic facts about the local differential geometry of rational homogeneous varieties. Mathematics Subject Classification (2000). 14M15, 20G05.

We connect the algebraic geometry and representation theory associated to Freudenthal's magic square. We give unified geometric descriptions of several classes of orbit closures, describing their hyperplane sections and desingularizations, and interpreting them in terms of composition algebras. In particular, we show how a class of invariant quartic polynomials can be viewed as generalizations of the classical discriminant of a cubic polynomial. ᮊ

We establish basic techniques for determining the ideals of secant varieties of Segre varieties. We solve a conjecture of Garcia, Stillman, and Sturmfels on the generators of the ideal of the first secant variety in the case of three factors and solve the conjecture set-theoretically for an arbitrary number of factors. We determine the low degree components of the ideals of secant varieties of small dimension in a few cases.

We study the quantum cohomology of (co)minuscule homogeneous varieties under a unified perspective. We show that three points Gromov-Witten invariants can always be interpreted as classical intersection numbers on auxiliary homogeneous varieties. Our main combinatorial tools are certain quivers, in terms of which we obtain a quantum Chevalley formula and a higher quantum Poincaré duality. In particular we compute the quantum cohomology of the two exceptional minuscule homogeneous varieties.

We give a computer-free proof of the Deligne, Cohen and de Man formulas for the dimensions of the irreducible g-modules appearing in g k ; k44; where g ranges over the exceptional complex simple Lie algebras. We give additional dimension formulas for the exceptional series, as well as uniform dimension formulas for other representations distinguished by Freudenthal along the rows of his magic chart. Our proofs use the triality model of the magic square, which we review and present a simplified proof of its validity. We conclude with some general remarks about obtaining ''series'' of Lie algebras in the spirit of Deligne and Vogel. # 2002 Elsevier Science (USA)

We discuss the geometry of orbit closures and the asymptotic behavior of Kronecker coefficients in the context of the Geometric Complexity Theory program to prove a variant of Valiant's algebraic analog of the P = N P conjecture. We also describe the precise separation of complexity classes that their program proposes to demonstrate.

We study, after Logachev, the geometry of smooth complex Fano threefolds X X with Picard number 1 1 , index 1 1 , and degree 10 10 , and their period map to the moduli space of 10-dimensional principally polarized abelian varieties. We prove that a general such X X has no nontrival automorphisms. By a simple deformation argument and a parameter count, we show that X X is not birational to a quartic double solid, disproving a conjecture of Tyurin. Through a detailed study of the variety of conics contained in X X , a smooth projective irreducible surface of general type with globally generated cotangent bundle, we construct two smooth projective two-dimensional components of the fiber of the period map through a general X X : one is isomorphic to the variety of conics in X X , modulo an involution, another is birationally isomorphic to a moduli space of semistable rank- 2 2 torsion-free sheaves on X X , modulo an involution. The threefolds corresponding to points of these components are obtained from X X via conic and line (birational) transformations. The general fiber of the period map is the disjoint union of an even number of smooth projective surfaces of this type.

We present a universal formula for the dimension of the Cartan powers of the adjoint representation of a complex simple Lie algebra (i.e., a universal formula for the Hilbert functions of homogeneous complex contact manifolds), as well as several other universal formulas. These formulas generalize formulas of Vogel and Deligne and are given in terms of rational functions where both the numerator and denominator decompose into products of linear factors with integer coefficients. We discuss consequences of the formulas including a relation with Scorza varieties.