12,474 research outputs found
A classification of Lagrangian planes in holomorphic symplectic varieties
Classically, an indecomposable class in the cone of effective curves on a
K3 surface is representable by a smooth rational curve if and only if
. We prove a higher-dimensional generalization conjectured by Hassett
and Tschinkel: for a holomorphic symplectic variety deformation equivalent
to a Hilbert scheme of points on a K3 surface, an extremal curve class
in the Mori cone is the line in a Lagrangian -plane
if and only if certain intersection-theoretic criteria
are met. In particular, any such class satisfies and the
primitive such classes are all contained in a single monodromy orbit.Comment: 18 pages, comments welcome. v3: classification extended to all curve
classes; some examples added. v4: to appear in J. Inst. Math. Jussie
Curves on Heisenberg invariant quartic surfaces in projective 3-space
This paper is about the family of smooth quartic surfaces that are invariant under the Heisenberg group . For a
very general such surface , we show that the Picard number of is 16 and
determine its Picard group. It turns out that the general Heisenberg invariant
quartic contains 320 smooth conics and that in the very general case, this
collection of conics generates the Picard group.Comment: Updated references, corrected typo
On tangents to quadric surfaces
We study the variety of common tangents for up to four quadric surfaces in
projective three-space, with particular regard to configurations of four
quadrics admitting a continuum of common tangents.
We formulate geometrical conditions in the projective space defined by all
complex quadric surfaces which express the fact that several quadrics are
tangent along a curve to one and the same quadric of rank at least three, and
called, for intuitive reasons: a basket. Lines in any ruling of the latter will
be common tangents.
These considerations are then restricted to spheres in Euclidean three-space,
and result in a complete answer to the question over the reals: ``When do four
spheres allow infinitely many common tangents?''.Comment: 50 page
How tangents solve algebraic equations, or a remarkable geometry of discriminant varieties
Let denote the discriminant variety of degree
polynomials in one variable with at least one of its roots being of
multiplicity . We prove that the tangent cones to
span thus, revealing an extreme ruled nature of these
varieties. The combinatorics of the web of affine tangent spaces to in is directly linked to the root multiplicities
of the relevant polynomials. In fact, solving a polynomial equation
turns out to be equivalent to finding hyperplanes through a given point
P(z)\in \mathcal D_{d,1} \approx \A^d which are tangent to the discriminant
hypersurface . We also connect the geometry of the Vi\`{e}te
map \mathcal V_d: \A^d_{root} \to \A^d_{coef}, given by the elementary
symmetric polynomials, with the tangents to the discriminant varieties
.
Various -partitions provide a refinement of the stratification of \A^d_{coef} by the 's. Our main result, Theorem 7.1, describes an intricate relation
between the divisibility of polynomials in one variable and the families of
spaces tangent to various strata .Comment: 43 pages, 12 figure
On organizing principles of Discrete Differential Geometry. Geometry of spheres
Discrete differential geometry aims to develop discrete equivalents of the
geometric notions and methods of classical differential geometry. In this
survey we discuss the following two fundamental Discretization Principles: the
transformation group principle (smooth geometric objects and their
discretizations are invariant with respect to the same transformation group)
and the consistency principle (discretizations of smooth parametrized
geometries can be extended to multidimensional consistent nets). The main
concrete geometric problem discussed in this survey is a discretization of
curvature line parametrized surfaces in Lie geometry. We find a discretization
of curvature line parametrization which unifies the circular and conical nets
by systematically applying the Discretization Principles.Comment: 57 pages, 18 figures; In the second version the terminology is
slightly changed and umbilic points are discusse
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