C ONTEMPORARY M ATHEMATICS 496 (cid:28)(cid:13)(cid:25)(cid:8)(cid:22)(cid:6)(cid:26)(cid:25)(cid:12)(cid:23)(cid:13)(cid:9)(cid:4)(cid:23)(cid:24)(cid:4)(cid:29)(cid:20)(cid:6)(cid:9)(cid:9)(cid:12)(cid:26)(cid:6)(cid:20)(cid:4) (cid:6)(cid:13)(cid:30)(cid:4)(cid:31) !(cid:8)(cid:22)(cid:12)(cid:26)(cid:6)(cid:20)(cid:4)"(cid:20)(cid:7)(cid:8)(cid:21)(cid:22)(cid:6)(cid:12)(cid:26)(cid:4) #(cid:8)(cid:23)!(cid:8)(cid:25)(cid:22)$ "(cid:4)(cid:29)(cid:23)(cid:13)(cid:24)(cid:8)(cid:22)(cid:8)(cid:13)(cid:26)(cid:8)(cid:4)(cid:12)(cid:13)(cid:4)%(cid:23)(cid:13)(cid:23)(cid:22)(cid:4)(cid:23)(cid:24)(cid:4)"(cid:13)(cid:30)(cid:22)(cid:8)&(cid:4)(cid:11)(cid:23)!!(cid:8)(cid:9)(cid:8) (cid:28)(cid:13)(cid:25)(cid:8)(cid:22)(cid:6)(cid:26)(cid:25)(cid:12)(cid:23)(cid:13)(cid:9)(cid:4)(cid:23)(cid:24)(cid:4)(cid:29)(cid:20)(cid:6)(cid:9)(cid:9)(cid:12)(cid:26)(cid:6)(cid:20)(cid:4)(cid:6)(cid:13)(cid:30)(cid:4)(cid:31) !(cid:8)(cid:22)(cid:12)(cid:26)(cid:6)(cid:20)(cid:4)"(cid:20)(cid:7)(cid:8)(cid:21)(cid:22)(cid:6)(cid:12)(cid:26)(cid:4)#(cid:8)(cid:23)!(cid:8)(cid:25)(cid:22)$(cid:4) ’(cid:6)$(cid:4)(()((cid:16)*(cid:4)((cid:19)(cid:19)+ ,(cid:13)(cid:12)(cid:27)(cid:8)(cid:22)(cid:9)(cid:12)(cid:25)$(cid:4)(cid:23)(cid:24)(cid:4)(cid:31)(cid:23)(cid:25)(cid:22)(cid:8)(cid:4)-(cid:6)!(cid:8) (cid:31)(cid:23)(cid:25)(cid:22)(cid:8)(cid:4)-(cid:6)!(cid:8)*(cid:4)(cid:28)(cid:13)(cid:30)(cid:12)(cid:6)(cid:13)(cid:6)(cid:4) -(cid:6)(cid:13)(cid:12)(cid:8)(cid:20)(cid:4)./(cid:4)0(cid:6)(cid:25)(cid:8)(cid:9)(cid:4) #(cid:12)(cid:6)(cid:13)’(cid:6)(cid:22)(cid:12)(cid:23)(cid:4)0(cid:8)(cid:9)(cid:6)(cid:13)(cid:6) (cid:11)(cid:6)(cid:13)(cid:30)(cid:22)(cid:6)(cid:4)-(cid:12)(cid:4)1(cid:23)(cid:26)(cid:26)(cid:23)(cid:4) (cid:29)2(cid:6)(cid:22)(cid:20)(cid:8)(cid:9)(cid:4)3/(cid:4)3(cid:6)!(cid:5)(cid:20)(cid:8)(cid:22) 4(cid:30)(cid:12)(cid:25)(cid:23)(cid:22)(cid:9) 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 Copying and reprinting. Materialinthisbookmaybereproducedbyanymeansforedu- cationaland scientific purposes without fee orpermissionwith the exception ofreproduction by servicesthatcollectfeesfordeliveryofdocumentsandprovidedthatthecustomaryacknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercialuseofmaterialshouldbeaddressedtotheAcquisitionsDepartment,AmericanMath- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can [email protected]. Excludedfromtheseprovisionsismaterialinarticlesforwhichtheauthorholdscopyright. In suchcases,requestsforpermissiontouseorreprintshouldbeaddresseddirectlytotheauthor(s). (Copyrightownershipisindicatedinthenoticeinthelowerright-handcornerofthefirstpageof eacharticle.) (cid:1)c 2009bytheAmericanMathematicalSociety. Allrightsreserved. TheAmericanMathematicalSocietyretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. Copyrightofindividualarticlesmayreverttothepublicdomain28years afterpublication. ContacttheAMSforcopyrightstatusofindividualarticles. PrintedintheUnitedStatesofAmerica. (cid:1)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://www.ams.org/ 10987654321 141312111009 To Andrew Sommese, collaborator, adviser, friend, on the occasion of his 60th birthday. This page intentionally left blank 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