Topics in Classical Algebraic Geometry

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1、TopicsinClassicalAlgebraicGeometryIGORV.DOLGACHEVJune28,2010iiContents1Polarity11.1Polarhypersurfaces...........................11.1.1Thepolarpairing........................11.1.2Thefirstpolars.........................41.1.3Thesecondpolars.......................61.1.4The

2、Hessianhypersurface...................81.1.5Parabolicpoints.........................111.1.6TheSteinerianhypersurface..................131.2Thedualhypersurface..........................161.2.1Thepolarmap.........................161.2.2Dualvarieties.......................

3、...171.2.3ThePluckerformulas¨......................201.3Polarpolyhedra.............................221.3.1Apolarschemes.........................221.3.2Sumsofpowers.........................241.3.3Generalizedpolarpolyhedra..................251.3.4Secantvarieties..........

4、...............261.3.5TheWaringproblems......................281.4Dualhomogeneousforms........................291.4.1Catalecticantmatrices.....................291.4.2Dualhomogeneousforms...................321.4.3TheWaringrankofahomogeneousform...........341.4.4Mukai’ssk

5、ew-symmetricform.................351.5Firstexamples..............................371.5.1Binaryforms..........................371.5.2Quadrics............................38Exercises...................................40HistoricalNotes................................422Coni

6、cs452.1Self-polartriangles...........................452.1.1TheVeronesequarticsurface..................452.1.2Polarlines...........................462.1.3Thevarietyofself-polartriangles...............47iiiivCONTENTS2.1.4Conjugatetriangles.......................502.2Ponc

7、eletrelation............................512.2.1Darboux’stheorem.......................512.2.2Ponceletcurves.........................562.2.3Invariantsofpairsofconics..................582.2.4TheSalmonconic.......................63Exercises.................................

8、..65HistoricalNotes................................673Planecubics693.1Equations...............................