1.1SupposeX is a plane curve andx ∈ X is its ordinary double point. Show that the pair consisting of the tangent line ofPa(X)atxand the lineaxis harmonically conjugate (see section2.1.2) to the pair of tangents to the branches ofX atxin the pencil of lines throughx. Ifxis an ordinary cusp, then show that the polar line of
Pa(X)atxis equal to the cuspidal tangent ofXatx.
1.2Show that a line contained in a hypersurfaceX belongs to all polars ofX with respect to any point on this line.
1.3Find the multiplicity of the intersection of a plane curveCwith its Hessian at an ordinary double point and at an ordinary cusp ofC. Show that the Hessian has a triple point at the cusp.
1.4Suppose a hypersurfaceX inPnhas a singular pointxof multiplicitym > 1. Prove that He(X)has this point as a point of multiplicity≥(n+ 1)m−2n.
1.5Suppose a hyperplane is tangent to a hypersurfaceXalong a closed subvarietyY
of codimension 1. Show thatY is contained in He(X).
1.6Supposef is the product ofddistinct linear formsli(t0, . . . , tn). LetA be the matrix of size(n+1)×dwhosei-th column is formed by the coefficients ofli(defined, of course up to proportionality). Let∆Ibe the maximal minor ofAcorresponding to a subsetIof[1, . . . , d]andfIbe the product of linear formsli, i6∈I. Show that
He(f) = (−1)n(d−1)fn−1X
I
∆2If 2 I
([397], p. 660).
1.7Find an example of a reduced hypersurface whose Hessian surface is nowhere re- duced.
1.8Show that the locus of points on the plane where the first polars of a plane curve
Xare tangent to each other is the Hessian ofXand the set of common tangents is the Cayleyan curve .
1.9Show that each inflection tangent of a plane curveX, considered as a point in the dual plane, lies on the Cayleyan ofX.
1.10Show that the class of the Steinerian St(X)of a plane curveXof degreedis equal to3(d−1)(d−2)but its dual is not equal to Cay(X).
1.11LetDm,n⊂Pmn−1be the image in the projective space of the variety ofm×n matrices of rank≤min{m, n} −1.
˜
Dm,n={(A, x)∈Pmn−1×Pn:A·x= 0}
is a resolution of singularities ofDm,n. Find the dual variety ofDm,n.1.12Find the dual variety of the Segre varietys(Pn×Pn)⊂Pn
2+2n
.
1.13LetXbe the union ofknonsingular conics in general position. Show thatX∨is also the union ofknonsingular conics in general position.
1.14LetXhas onlyδordinary nodes andκordinary cusps as singularities. Assume that the dual curveX∨has also onlyδˇordinary nodes andκˇordinary cusps as singularities. Findδˇandκˇin terms ofd, δ, κ.
1.15Give an example of a self-dual (i.e.X∨∼=X) plane curve of degree>2.
1.16Show that the Jacobian of a net of plane curves has a double point at each simple base point unless the net contains a curve with a triple point at the base point [215].
1.17Let|L|be a generaln-dimensional linear system of quadrics inPnand|L|⊥be the( n+22
−n−2)-dimensional subspace of apolar quadric in the dual space. Show that the variety of reducible quadrics in|L|⊥is isomorphic to the Reye variety of|L|
and has the same degree.
1.18Show that the embedded tangent space of the Veronese varietyVndat a point repre- sented by the formldis equal to the projectivization of the linear space of homogeneous polynomials of degreedof the formld−1m.
1.19Using the following steps, show thatV34is6-defective by proving that for 7 general pointspiinP4there is a cubic hypersurface with singular points at thepi’s.
(i) Show that there exists a Veronese curveR4of degree 4 through the seven points. (ii) Show that the secant variety ofR4 is a cubic hypersurface which is singular
alongR4.
1.20Letqbe a nondegenerate quadratic form inn+ 1variables. Show that VSP(q, n+ 1)oembedded inG(n, E)is contained in the linear subspace of codimensionn.
1.21Compute the catalecticant matrix Cat2(f), where f is a homogeneous form of degree 4 in 3 variables.
1.22Letf∈S2k(E∨
)andΩf be the corresponding quadratic form onSk(E). Show that the quadricV(Ωf)in|Sk(E)|is characterized by the following two properties:
• Ωf is apolar to any quadric in|Sk(E∨)|which contains the image of the Veronese map|E∨|=P(E)→ |Sk(E∨)|=P(Sk(E)).
1.23LetCkbe the locus in|S2k(E∨)|of hypersurfacesV(f)such thatdetCatk(f) =
0. Show thatCkis a rational variety. [Hint: Consider the rational mapCk 99K |E|) which assigns toV(f)the point defined by the subspace APk(f)and study its fibres].
1.24Give an example of a polar 4-gon of the cubict0t1t2= 0.
1.25Find all binary forms of degreedfor which VSP(f,2)o=∅.
1.26Letfbe a form of degreedinn+ 1variables. Show that VSP(f, n+d d
)o
is an irreducible variety of dimensionn n+dd
.
1.27Describe the variety VSP(f,4), wherefis a nondegenerate quadratic form in 3 variables.
1.28Show that a smooth pointyof a hypersurfaceX belongs to the intersection of the polar hypersurfacesPx(X)andPx2(X)if and only if the line connectingxandy
intersectsXat the pointywith multiplicity≥3.
1.29Show that the vertices of two polar tetrahedra of a nonsingular quadric inP3are base points of a net of quadrics. Conversely, the set of 8 base points of a general net of quadrics can be divided in any way into two sets, each of two sets is the set of vertices of a polar tetrahedron of the same quadric[538].
1.30Suppose two cubic plane curvesV(f)andV(g)admit a common polar pentagon. Show that the determinant of the6×6-matrix[Cat1(f)Cat1(g)]vanishes [225].