5,563 research outputs found
Singularities of plane rational curves via projections
We consider the parameterization f=(f0:f1:f2)of a plane rational curve C of degree n, and we study the singularities of C via such parameterization. We use the projection from the rational normal curve CnâPn to C and its interplay with the secant varieties to Cn. In particular, we define via f certain 0-dimensional schemes XkâPk, 2â€kâ€(nâ1), which encode all information on the singularities of multiplicity â„k of C (e.g. using X2 we can give a criterion to determine whether C is a cuspidal curve or has only ordinary singularities). We give a series of algorithms which allow one to obtain information about the singularities from such schemes
Scrolls and hyperbolicity
Using degeneration to scrolls, we give an easy proof of non-existence of
curves of low genera on general surfaces in P3 of degree d >=5. We show, along
the same lines, boundedness of families of curves of small enough genera on
general surfaces in P3. We also show that there exist Kobayashi hyperbolic
surfaces in P3 of degree d = 7 (a result so far unknown), and give a new
construction of such surfaces of degree d = 6. Finally we provide some new
lower bounds for geometric genera of surfaces lying on general hypersurfaces of
degree 3d > 15 in P4.Comment: 17
Singular Curves of Low Degree and Multifiltrations from Osculating Spaces
In order to study projections of smooth curves, we introduce multifiltrations
obtained by combining flags of osculating spaces. We classify all
configurations of singularities occurring for a projection of a smooth curve
embedded by a complete linear system away from a projective linear space of
dimension at most two. In particular, we determine all configurations of
singularities of non-degenerate degree d rational curves in when
and . Along the way, we describe the Schubert cycles
giving rise to these projections.
We also reprove a special case of the Castelnuovo bound using these
multifiltrations: under the assumption , the arithmetic genus of any
nondegenerate degree curve in is at most .Comment: 34 pages, 11 tables, 2 figures; v2 added references and made minor
corrections; v3 more minor revisions, to appear in IMR
Equivalent birational embeddings II: divisors
Two divisors in are said to be Cremona equivalent if there is a
Cremona modification sending one to the other. We produce infinitely many non
equivalent divisorial embeddings of any variety of dimension at most 14. Then
we study the special case of plane curves and rational hypersurfaces. For the
latter we characterise surfaces Cremona equivalent to a plane.Comment: v2 Exposition improved, thanks to referee, unconditional
characterization of surfaces Cremona equivalent to a plan
Changing Views on Curves and Surfaces
Visual events in computer vision are studied from the perspective of
algebraic geometry. Given a sufficiently general curve or surface in 3-space,
we consider the image or contour curve that arises by projecting from a
viewpoint. Qualitative changes in that curve occur when the viewpoint crosses
the visual event surface. We examine the components of this ruled surface, and
observe that these coincide with the iterated singular loci of the coisotropic
hypersurfaces associated with the original curve or surface. We derive
formulas, due to Salmon and Petitjean, for the degrees of these surfaces, and
show how to compute exact representations for all visual event surfaces using
algebraic methods.Comment: 31 page
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