295 research outputs found

### Real Rational Curves in Grassmannians

Fulton asked how many solutions to a problem of enumerative geometry can be
real, when that problem is one of counting geometric figures of some kind
having specified position with respect to some general fixed figures. For the
problem of plane conics tangent to five general conics, the (surprising) answer
is that all 3264 may be real. Similarly, given any problem of enumerating
p-planes incident on some general fixed subspaces, there are real fixed
subspaces such that each of the (finitely many) incident p-planes are real. We
show that the problem of enumerating parameterized rational curves in a
Grassmannian satisfying simple (codimension 1) conditions may have all of its
solutions be real.Comment: 9 pages, 1 eps figure, uses epsf.sty. Below the LaTeX source is a
MAPLE V.5 file which computes an example in the paper, and its outpu

### Some real and unreal enumerative geometry for flag manifolds

We present a general method for constructing real solutions to some problems
in enumerative geometry which gives lower bounds on the maximum number of real
solutions. We apply this method to show that two new classes of enumerative
geometric problems on flag manifolds may have all their solutions be real and
modify this method to show that another class may have no real solutions, which
is a new phenomenon. This method originated in a numerical homotopy
continuation algorithm adapted to the special Schubert calculus on
Grassmannians and in principle gives optimal numerical homotopy algorithms for
finding explicit solutions to these other enumerative problems.Comment: 19 pages, LaTeX-2e; Updated and final version. To appear in the issue
of Michigan Mathematical Journal dedicated to Bill Fulto

### An excursion from enumerative goemetry to solving systems of polynomial equations with Macaulay 2

Solving a system of polynomial equations is a ubiquitous problem in the
applications of mathematics. Until recently, it has been hopeless to find
explicit solutions to such systems, and mathematics has instead developed deep
and powerful theories about the solutions to polynomial equations. Enumerative
Geometry is concerned with counting the number of solutions when the
polynomials come from a geometric situation and Intersection Theory gives
methods to accomplish the enumeration.
We use Macaulay 2 to investigate some problems from enumerative geometry,
illustrating some applications of symbolic computation to this important
problem of solving systems of polynomial equations. Besides enumerating
solutions to the resulting polynomial systems, which include overdetermined,
deficient, and improper systems, we address the important question of real
solutions to these geometric problems.
The text contains evaluated Macaulay 2 code to illuminate the discussion.
This is a chapter in the forthcoming book "Computations in Algebraic Geometry
with Macaulay 2", edited by D. Eisenbud, D. Grayson, M. Stillman, and B.
Sturmfels. While this chapter is largely expository, the results in the last
section concerning lines tangent to quadrics are new.Comment: LaTeX 2e, 22 pages, 1 .eps figure. Source file (.tar.gz) includes
Macaulay 2 code in article, as well as Macaulay 2 package realroots.m2
Macaulay 2 available at http://www.math.uiuc.edu/Macaulay2 Revised with
improved exposition, references updated, Macaulay 2 code rewritten and
commente

### Enumerative geometry for real varieties

We discuss the problem of whether a given problem in enumerative geometry can
have all of its solutions be real. In particular, we describe an approach to
problems of this type, and show how this can be used to show some enumerative
problems involving the Schubert calculus on Grassmannians may have all of their
solutions be real. We conclude by describing the work of Fulton and
Ronga-Tognoli-Vust, who (independently) showed that there are 5 real plane
conics such that each of the 3264 conics tangent to all five are real.Comment: Based upon the Author's talk at 1995 AMS Summer Research Institute in
Algebraic geometry. To appear in the Proceedings. 11 pages, extended version
with Postscript figures and appendix available at
http://www.msri.org/members/bio/sottile.html, or by request from Author
([email protected]

### Pieri-type formulas for maximal isotropic Grassmannians via triple intersections

We give an elementary proof of the Pieri-type formula in the cohomology of a
Grassmannian of maximal isotropic subspaces of an odd orthogonal or symplectic
vector space. This proof proceeds by explicitly computing a triple intersection
of Schubert varieties. The decisive step is an explicit description of the
intersection of two Schubert varieties, from which the multiplicities (which
are powers of 2) in the Pieri-type formula are deduced.Comment: LaTeX 2e, 24 pages (9 pages is an appendix detailing the proof in the
symplectic case). Expanded version of MSRI preprint 1997-06

### Real enumerative geometry and effective algebraic equivalence

We describe an approach to the question of finding real solutions to problems
of enumerative geometry, in particular the question of whether a problem of
enumerative geometry can have all of its solutions be real. We give some
methods to infer one problem can have all of its solutions be real, given that
a related problem does. These are used to show many Schubert-type enumerative
problems on some flag manifolds can have all of their solutions be real.Comment: 12 pages, LaTeX 2

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