This paper proves three statements of Schubert about cuspal cubic curves in a plane by using the concept of generic point of Van der Waerden and Weil and Ritt-Wu methods.They are relations of some special lines:1)For ...This paper proves three statements of Schubert about cuspal cubic curves in a plane by using the concept of generic point of Van der Waerden and Weil and Ritt-Wu methods.They are relations of some special lines:1)For a given point,all the curves containing this point are considered.For any such curve,there are five lines.Two of them are the tangent lines of the curve passing through the given point.The other three are the lines connecting the given point with the cusp,the inflexion point and the intersection point of the tangent line at the cusp and the inflexion line.2)For a given point,the curves whose tangent line at the cusp passes through this point are considered.For any such curve,there are four lines.Three of them are the tangent lines passing through this point and the other is the line connect the given point and the inflexion point.3)For a given point,the curves whose cusp,inflexion point and the given point are collinear are considered.For any such curve,there are five lines.Three of them are tangent lines passing through the given point.The other two are the lines connecting the given point with the cusp and the intersection point of the tangent line at the cusp and the inflexion line.展开更多
Hilbert problem 15 requires to understand Schubert's book. In this book, there is a theorem in §23, about the relation of the tangent lines from a point and the singular points of cubed curves with cusp near ...Hilbert problem 15 requires to understand Schubert's book. In this book, there is a theorem in §23, about the relation of the tangent lines from a point and the singular points of cubed curves with cusp near a 3-multiple straight line, which was obtained by the so called main trunk numbers, while for these numbers, Schubert said that he obtained them by experiences. So essentially Schubert even did not give any hint for the proof of this theorem. In this paper, by using the concept of generic point in the framework of Van der Waerden and Weil on algebraic geometry, and realizing Ritt-Wu method on computer, the authors prove that this theorem of Schubert is completely right.展开更多
In this paper,we give rigorous justification of the ideas put forward in§20,Chapter 4 of Schubert’s book;a section that deals with the enumeration of conics in space.In that section,Schubert introduced two degen...In this paper,we give rigorous justification of the ideas put forward in§20,Chapter 4 of Schubert’s book;a section that deals with the enumeration of conics in space.In that section,Schubert introduced two degenerate conditions about conics,i.e.,the double line and the two intersection lines.Using these two degenerate conditions,he obtained all relations regarding the following three conditions:conics whose planes pass through a given point,conics intersecting with a given line,and conics which are tangent to a given plane.We use the language of blow-ups to rigorously treat the two degenerate conditions and prove all formulas about degenerate conditions stemming from Schubert’s idea.展开更多
Hilbert Problem 15 required an understanding of Schubert’s book[1],both its methods and its results.In this paper,following his idea,we prove that the formulas in§6,§7,§10,about the incidence of points...Hilbert Problem 15 required an understanding of Schubert’s book[1],both its methods and its results.In this paper,following his idea,we prove that the formulas in§6,§7,§10,about the incidence of points,lines and planes,are all correct.As an application,we prove formulas 8 and 9 in§12,which are frequently used in his book.展开更多
In§13 of Schubert’s famous book on enumerative geometry,he provided a few formulas called coincidence formulas,which deal with coincidence points where a pair of points coincide.These formulas play an important ...In§13 of Schubert’s famous book on enumerative geometry,he provided a few formulas called coincidence formulas,which deal with coincidence points where a pair of points coincide.These formulas play an important role in his method.As an application,Schubert utilized these formulas to give a second method for calculating the number of planar curves in a one dimensional system that are tangent to a given planar curve.In this paper,we give proofs for these formulas and justify his application to planar curves in the language of modern algebraic geometry.We also prove that curves that are tangent to a given planar curve is actually a condition in the space of planar curves and other relevant issues.展开更多
文摘This paper proves three statements of Schubert about cuspal cubic curves in a plane by using the concept of generic point of Van der Waerden and Weil and Ritt-Wu methods.They are relations of some special lines:1)For a given point,all the curves containing this point are considered.For any such curve,there are five lines.Two of them are the tangent lines of the curve passing through the given point.The other three are the lines connecting the given point with the cusp,the inflexion point and the intersection point of the tangent line at the cusp and the inflexion line.2)For a given point,the curves whose tangent line at the cusp passes through this point are considered.For any such curve,there are four lines.Three of them are the tangent lines passing through this point and the other is the line connect the given point and the inflexion point.3)For a given point,the curves whose cusp,inflexion point and the given point are collinear are considered.For any such curve,there are five lines.Three of them are tangent lines passing through the given point.The other two are the lines connecting the given point with the cusp and the intersection point of the tangent line at the cusp and the inflexion line.
文摘Hilbert problem 15 requires to understand Schubert's book. In this book, there is a theorem in §23, about the relation of the tangent lines from a point and the singular points of cubed curves with cusp near a 3-multiple straight line, which was obtained by the so called main trunk numbers, while for these numbers, Schubert said that he obtained them by experiences. So essentially Schubert even did not give any hint for the proof of this theorem. In this paper, by using the concept of generic point in the framework of Van der Waerden and Weil on algebraic geometry, and realizing Ritt-Wu method on computer, the authors prove that this theorem of Schubert is completely right.
基金partially supported by National Center for Mathematics and Interdisciplinary Sciences,CAS。
文摘In this paper,we give rigorous justification of the ideas put forward in§20,Chapter 4 of Schubert’s book;a section that deals with the enumeration of conics in space.In that section,Schubert introduced two degenerate conditions about conics,i.e.,the double line and the two intersection lines.Using these two degenerate conditions,he obtained all relations regarding the following three conditions:conics whose planes pass through a given point,conics intersecting with a given line,and conics which are tangent to a given plane.We use the language of blow-ups to rigorously treat the two degenerate conditions and prove all formulas about degenerate conditions stemming from Schubert’s idea.
基金partially supported by National Center for Mathematics and Interdisciplinary Sciences,CAS。
文摘Hilbert Problem 15 required an understanding of Schubert’s book[1],both its methods and its results.In this paper,following his idea,we prove that the formulas in§6,§7,§10,about the incidence of points,lines and planes,are all correct.As an application,we prove formulas 8 and 9 in§12,which are frequently used in his book.
基金supported by National Center for Mathematics and Interdisciplinary Sciences,CAS。
文摘In§13 of Schubert’s famous book on enumerative geometry,he provided a few formulas called coincidence formulas,which deal with coincidence points where a pair of points coincide.These formulas play an important role in his method.As an application,Schubert utilized these formulas to give a second method for calculating the number of planar curves in a one dimensional system that are tangent to a given planar curve.In this paper,we give proofs for these formulas and justify his application to planar curves in the language of modern algebraic geometry.We also prove that curves that are tangent to a given planar curve is actually a condition in the space of planar curves and other relevant issues.
