Interactions of Classical and Numerical Algebraic Geometry: A Conference in Honor of Andrew Sommese, Interactions of Classical and Numerical Algebraic ... Dame, Notre D

378 Pages·2009·4.12 MB·English

by  Daniel J. Bates

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American Mathematical Society
Interactions of Classical
and Numerical Algebraic
Geometry
This page intentionally left blank
C
ONTEMPORARY
M
ATHEMATICS
496
Interactions of Classical
and Numerical Algebraic
Geometry
A Conference in Honor of Andrew Sommese
Interactions of Classical and Numerical Algebraic Geometry
May 22–24, 2008
University of Notre Dame
Notre Dame, Indiana
Daniel J. Bates
GianMario Besana
Sandra Di Rocco
Charles W. Wampler
Editors
American Mathematical Society
Providence, Rhode Island
Editorial Board
Dennis DeTurck, managing editor
George Andrews Abel Klein Martin J. Strauss
2000 Mathematics Subject Classification. Primary13Pxx, 14C17, 14C20, 14C25, 14D06,
14D20, 14Hxx, 14Jxx, 14Mxx, 14Q15, 65H10, 65H20.
Anyopinions,findings,andconclusionsorrecommendationsexpressedinthismaterial
arethoseoftheauthor(s)anddonotnecessarilyreflecttheviewsoftheNationalScience
Foundation.
Library of Congress Cataloging-in-Publication Data
Interactionsofclassicalandnumericalalgebraicgeometry: aconferenceinhonorofA.J.Sommese,
May 22–24, 2008, University of Notre Dame, Notre Dame, Indiana / Daniel J. Bates...[et al.],
editors.
p.cm. —(Contemporarymathematics;v.496)
Includesbibliographicalreferences.
ISBN978-0-8218-4746-6
1.Geometry,Algebraic—Congresses. I.Sommese,AndrewJohn. II.Bates,DanielJ.(Daniel
James),1979–
QA564.I56 2009
516.3(cid:1)5—dc22 2009011259
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(cid:1)c 2009bytheAmericanMathematicalSociety. Allrightsreserved.
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Copyrightofindividualarticlesmayreverttothepublicdomain28years
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establishedtoensurepermanenceanddurability.
VisittheAMShomepageathttp://www.ams.org/
10987654321 141312111009
To Andrew Sommese, collaborator, adviser, friend,
on the occasion of his 60th birthday.
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Contents
Preface ix
Turning points and bifurcations for homotopies of analytic maps
E. Allgower, S.-G. Cruceanu, and S. J. Tavener 1
On a connectedness theorem of Debarre
L. Ba˘descu 11
Stepsize control for path tracking
D. J. Bates, J. D. Hauenstein, A. J. Sommese,
and C. W. Wampler II 21
A primer on Seshadri constants
T. Bauer, S. Di Rocco, B. Harbourne, M. Kapustka, A. Knutsen,
W. Syzdek, and T. Szemberg 33
A view on extending morphisms from ample divisors
M. C. Beltrametti and P. Ionescu 71
Isotropic models of evolution with symmetries
W. Buczyn´ska, M. Donten, and J. A. Wi´sniewski 111
Special scrolls whose base curve has general moduli
A. Calabri, C. Ciliberto, F. Flamini, and R. Miranda 133
On varieties whose universal cover is a product of curves (with an appendix
by A. J. Di Scala)
F. Catanese and M. Franciosi 157
Ideals of numeric realizations of configurations of lines
B. H. Dayton 181
The standard filtration on cohomology with compact supports with an
appendix on the base change map and the Lefschetz hyperplane theorem
M. A. A. de Cataldo 199
A family of exceptional Stewart Gough mechanisms of genus 7
F. Geiss and F.-O. Schreyer 221
Numerical computation of the dimensions of the cohomology of twists of ideal
sheaves
J. D. Hauenstein, J. C. Migliore, C. Peterson,
and A. J. Sommese 235
vii
viii CONTENTS
Polyhedral methods in numerical algebraic geometry
J. Verschelde 243
Double covers of Del Pezzo manifolds and bielliptic curve sections
A. Lanteri and H. Maeda 265
Total degree vs. mixed volume
T. Y. Li, X. Wang, and Y. Zhang 283
Varieties swept out by Grassmanians of lines
R. Mun˜oz and L. E. Sola´ Conde 303
Submanifolds with ample normal bundles and a conjecture of Hartshorne
T. Peternell 317
A complete kinematic analysis of the SNU 3-UPU parallel robot
D. R. Walter, M. L. Husty, and M. Pfurner 331
The closedness subspace method for computing the multiplicity structure of a
polynomial system
Z. Zeng 347
Preface
Algebraic geometry is a classical discipline which for many years sat at the
intersection of algebra, number theory, several complex variables, and geometry
in all its incarnations. The advent of personal computing, and more so the de-
velopment of software for symbolic computations, introduced a new facet of the
disciplineinthe1980’s. SoftwarepackagessuchasSingular,Macaulay,andCoCoA
have partially changed the scope of problems that can be considered. Nonetheless,
the mainstream approach of algebraic geometry remained distinctly separate from
the computational (particularly numerical) flavor found in many other branches of
mathematics.
In the 1990’s, a numerical approach to algebraic geometry was pioneered, due
largely to the work of Andrew Sommese and his collaborators. In the early days,
the fundamental technique, homotopy continuation, was used to compute solu-
tions to polynomial systems arising mostly from kinematics and, more generally,
engineering. These new numerical techniques opened the door for researchers to
attack polynomial systems from a different angle, in many cases expanding the
classes of polynomial systems that could be solved in practice. In the last few
years, the numerical approach has grown quickly and extended its reach into areas
often considered to be strictly the domain of exact symbolic computation. It is
nowbeingrealizedthatnumerical techniquesandsymbolic, classical methodsneed
not compete and can harmoniously complement each other, as multiple tools in
the practitioner’s toolbox. Active sharing of ideas, progress, and problems in both
directions between the classical and numerical algebraic geometry communities is
vital.
Accordingly,withthehelpofJuanMigliore,weorganizedaconferenceaimedat
enhancing this interaction. The conference was titled Interactions of Classical and
NumericalAlgebraicGeometryandtookplaceattheUniversityofNotreDameover
a period of three days in May 2008. There were 11 talks by world leaders in these
twofields, withalivelydiscussionperiodattheendofeachday. Believingthatthe
intersectionofthetwofieldswas(andis)ripeforrapidgrowthfueledbyjointwork,
we hoped that bringing these communities together in an interactive forum would
help spark important advances. The strong participation in the meeting indicates
that the communities are indeed receptive to interaction, and the contributions to
this volume show that valuable results have already begun to accrue.
The inspiration of this meeting was to honor the career, past and present, of
a pioneer of both classical and particularly numerical algebraic geometry, Profes-
sorAndrewJ.Sommese,theVincentJ.andAnnamarieMicusDuncanProfessorof
MathematicsattheUniversityofNotreDame,intheyearofhis60thbirthday. An-
drew spent roughly the first 15-20 years of his career primarily focused on complex
ix