polar and dual varieties of real curves and surfaces

Reciprocal and dual varieties Polar varieties Curves and surfaces Reciprocal polars

Polar and dual varietiesof real curves and surfaces

Heidi Mork and Ragni PieneCentre of Mathematics for Applications,

University of Oslo, Norway

IMA, September 19, 2006


Outline

Reciprocal and dual varieties

Polar varieties

Curves and surfaces

Reciprocal polars


Motivation

Algebraic geometry: study the common solutions of a given set ofpolynomial equations.

In particular, what can one say about the set of real solutions? Thenumber of connected components? The shape of each component?

For practical purposes it is often useful to locate explicitly a pointon each component, and then proceed from there.

Bank, Giusti, Heintz, Mbakop, Pardo, and Safey El Din andSchost, used classical algebraic geometry — the theory of polarvarieties — to describe an algorithm which produces a point oneach real component of a smooth variety.


Reciprocal varieties

Recall the notion of “poles and polars”:

If Q ⊂ Pn is a non-degenerate quadric, then

I if P is a point, its polar hyperplane P⊥ with respect to Q isthe linear span of the points on Q such that the tangenthyperplane to Q at that point passes through P . This is thesame as the hyperplane

∑ ∂Q∂Xi

(P )Xi = 0.

I if H is a hyperplane, then its polar point H⊥ is theintersection of the tangent hyperplanes to Q at the points onQ ∩H.

I if L is a linear space, its polar space L⊥ is the intersection ofthe polar hyperplanes to points in L.

If the quadric is Q :∑

X2i = 0, then the polar of a point

P = (p0 : . . . : pn) is the hyperplane p0X0 + . . . pnXn = 0.


Let X ⊂ Pn be a hypersurface. Construct the “reciprocal” varietyX∗ ⊂ Pn as follows:

To each point P ∈ X, take the tangent hyperplane TX,P , and takethe polar point P ∗ := T⊥

X,P of this hyperplane with respect to thequadric Q.

The reciprocal variety:

X∗ = {P ∗|P ∈ X} ⊂ Pn

(Dieudonne: Grassmann’s greatest tragedy was that he did notmake a distinction between a projective space and the dualprojective space . . . .)


Dual varieties

The dual projective space: Pn∨ = {H|H ⊂ Pn hyperplane }

The dual variety of X ⊂ Pn is

X∨ = {H ∈ Pn∨|H ⊇ TX,P , P ∈ X}

The (projective) Gauss map is the (rational) map

γ:X → Grass(r, n)

given by γ(P ) = TX,P , where r = dim X.

If X is a hypersurface, we have X∗ = Q(γ(X)), whereγ:X → Grass(n− 1, n) = Pn∨ and Q: Pn∨ → Pn is the polaritywith respect to Q.


The dual surfaces of Steiner surfaces:

A, B: The dual surface is a cubic surface with 4 A1 singularitiesand containing 9 lines.

D: If the surface has equation

w4 − 2w2x(y + z) + x2(x− z)2 = 0.

then the dual surface is the cubic

4stu− v2(s + t) = 0

which has two A1 singularities and one A3 singularity and containsfive lines.


F: The nodal curve is a higher order tacnodal line (A5). We mayassume the surface has equation

(wy − z2)2 − xy3 = 0.

The dual surface is the cubic

s2u− 4tuv + v3 = 0.

This cubic surface has one A1 and one A5 singularity, and itcontains two lines.


Polar varieties

The polar varieties Mk of a projective variety X ⊂ Pn wereintroduced by Severi (for curves), and Todd, who introduced thename and used them to:

I define characteristic classes

ck(X) =k∑

i=0

(−1)k−i

(r + 1− k + i

i

)[Mk]hi

where r is the dimension of X and h is the class of ahyperplane section of X.

Example: if X is a plane curve (r = 1, n = 2), then the Eulernumber e(X) = c1(X) of X is∫

(−[M1] + 2h) = −d(d− 1) + 2d = d(3− d) = 2− 2g.


Consider a flag L0 ⊂ . . . ⊂ Ln−1 ⊂ Pn and (for simplicity) ahypersurface X ⊂ Pn.

The kth polar variety of X is

Mk = {P ∈ X|TX,P ⊇ Lk−1}.

There is a natural scheme structure on Mk (Fitting ideals), and byKleiman’s transversality theorem, for a general flag they are “nice.”

One can define (by restriction) affine polar varities, and one candefine real projective and affine polar varieties, and X can besingular.


Geometric interpretation

We have M1 = γ−1(L∨0 ), where γ:X → Pn∨ is the Gauss map.

The first polar variety M1 is the ramification locus of theprojection of X from the point L0.

The real part, M1(R), is the apparent contour of X(R) viewedfrom L0.

The contour of a connected compact component of X(R) iscompact. If a component is non-compact, the contour on thatcomponent could be empty.

M1(R) does not meet every connected component of X(R) if andonly if the hyperplane L∨

0 ⊂ Pn∨ does not intersect all connectedcomponents of X∨(R).


Bank et al. showed: If X(R) is smooth and compact, then ageneral polar variety has a real point on each connectedcomponent of X(R).

I What if X is singular?

I The dual variety of a smooth variety is (almost) alwayssingular.

I X(R) compact does not imply X∨(R) is compact.

For example, a plane curve C(R) which consists of only ovals withno singularities and no inflections, has compact dual C(R)∨ iff theovals have a common interior point.


Curves

Use polar varieties to bound the number of real components?The number for a smooth curve of even degree is at mostd(d− 1)/2 — but this is not as good as Harnack.

Extend Bank et al. to the singular case?

Can show: Let C be a plane curve such that C(R) is compact andwith only ordinary multiple points as singularities. Then M1(R)contains a smooth point on each circuit of C(R).

What about the non-compact case?


The double pillow

x4 − 2x2y2 + y4 − 2x2w2 − 2y2w2 − 16z2w2 + w4 = 0

Picture in xyz-space (w = 1):

x4 − 2x2y2 + y4 − 2x2 − 2y2 − 16z2 + 1 = 0


Picture in xyw-space (z = 1):

x4 − 2x2y2 + y4 − 2x2w2 − 2y2w2 − 16w2 + w4 = 0


The dual surface of the double pillow:

16x2 − y4 + 2y2z2 − 8x2y2 − z4 − 8x2z2 − 16x4 = 0

Most planes will not meet this surface. Hence for most points, thepolar variety M1(R) of the double pillow will be empty.


The shape of surfaces

To get a quick understanding of the shape of a given surface ofdegree d, make its equation f(x, y, z) to be monic in z. Computethe resultant R0(x, y) and certain subdeterminants Ri of theresultant matrix, i = 1, ..., d− 2. These are the Sturm–Habichtcoefficients.

The arrangement of the curves Ri = 0 in the wanted region in thexy-plane gives regions (surface patches, curves, and points) wherethe number of real solutions in z to f(x, y, z) = 0 is constant.The curve R0(x, y) = 0 is the projection of the contour of thesurface to the (x, y)-plane. It is also the projection of the firstpolar variety.


Example: the double pillow

In the w = 1 case: The arrangement is given by the curve R0 = 0,which is

x4 − 2x2y2 + y4 − 2x2 − 2y2 + 1 = 0

This gives the four lines ±x± y = 1.For points (x, y) on the lines, there is one solution in w, for pointsin the square and in the four cones, there are two solutions, and forpoints in the rest of the plane there are no real solutions.

In the z = 1 case: R0 = 0 gives the lines x + y = 0 and x− y = 0,whereas neither R1 = 0 nor R2 = 0 has real solutions.For points (x, y) on the lines there are three real solutions in w, forpoints outside the lines there are four.


Reciprocal polar varieties

Consider again a flag L0 ⊂ . . . ⊂ Ln−1 ⊂ Pn, a quadric Q, and ahypersurface X ⊂ Pn. Take the reciprocal flag:

L⊥n−1 ⊂ . . . ⊂ L⊥

0 ⊂ Pn

and define the kth reciprocal polar variety

M⊥k = {P ∈ X|TX,P ⊇ 〈P,L⊥

n−k〉⊥}

= {P ∈ X|T⊥X,P ⊆ 〈P,L⊥

n−k〉}

In particular,

M1 = {P ∈ X|L⊥n−1 ∈ 〈T⊥

X,P , P 〉}


M⊥1 (R) is the locus of points on X(R) closest to L∨

n−1, withrespect to the metric defined by Q, since 〈T⊥

X,P , P 〉 is the normalto X at P .

If X is a curve, given by f(x, y, z) = 0, then the equation of thefirst polar with respect to L0 = (a : b : c) is

a∂f

∂x+ b

∂f

∂y+ c

∂f

∂z= 0

and, if Q:x2 + y2 + z2 = 0, then the first “reciprocal” polar withrespect to the line L1: a′x + b′y + c′z = 0 is

(c′y − b′z)∂f

∂x− (c′x− a′z)

∂f

∂y+ (b′x− a′y)

∂f

∂z= 0.


Hence the degree of [M1] is d(d− 1), and the degree of [M⊥1 ] is d2.

Hence it should be “easier” for the reciprocal polars to meet eachreal component of X than for the usual polars — and indeed:

Bank et al. showed: If X is a complete intersection variety suchthat X(R) is smooth, then any general reciprocal polar variety willmeet every connected component of X(R).

polar and dual varieties of real curves and surfaces

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