Propuesta de políticas de Gestión de Proyecto

Let us assume that the determinant representation of a plane irreducible curve

Cof degreedis given by a pair of equal invertible sheavesL=M. It follows from Lemmas4.1.1and4.1.3that

• L⊗2_{∼=OC(d−1);} • deg(L) =1₂d(d−1); • H0_{(C,L(−1)) ={0}.}

Recall that the canonical sheafωCis isomorphic toOC(d−3). Thus

L(−1)⊗2∼=ωC. (4.7)

Definition 4.1.6 LetX be a curve with a canonical invertible sheafωX(e.g. a nonsingular curve, or a curve on a nonsingular surface). An invertible sheaf

θwhose tensor square is isomorphic toωX is called atheta characteristic. A theta characteristic is calledeven(resp.odd) ifdim H0_{(X,N)is even (resp.} odd).

Using this definition, we can express (4.7) by saying that L ∼=θ(1),

whereθis an even theta characteristic (becauseH0_{(C, θ) = {0}). Of course,} the latter condition is stronger. An even theta characteristic with no nonzero global sections (resp. with nonzero global sections) is called a non-effective theta characteristic(resp.effective theta characteristic).

Rewriting the previous Subsection under the assumption thatL = M, we obtain thatU = V. The mapsl =rare given by the linear systems|L|and define a map(l,l) :C →P(U)×P(U). Its composition with the Segre map P(U)×P(U)→P(U⊗U)and the projection toP(S2(U∨))defines a map

In coordinates, the map is given by

ψ(x) = ˜l(x)·t_˜l(x),

where˜l(x)is the column of projective coordinates of the pointl(x). It is clear that the image of the mapψis contained in the variety of rank 1 quadrics in |U∨|. It follows from the proof of Theorem4.1.4that there exists a linear map

φ : P2 → |S2(U∨)|such that its composition with the rational map defined

by taking the adjugate matrix is equal, after restriction toC, to the map ψ. The image ofφis a netN of quadrics in|U|. The imageφ(C)is the locus of singular quadrics inN. For each pointx∈ C, we denote the corresponding quadric byQx. The regular maplis defined by assigning to a pointx∈Cthe singular point of the quadricQx. The imageXofCin|U|is a curve of degree equal todegL=1₂d(d−1).

Proposition 4.1.7 The restriction map

r:H0(|U|,O|U|(2))→H0(X,OX(2)) is bijective. Under the isomorphism

H0(X,OX(2))∼=H0(C,L⊗2_)∼

=H0(C,OC(d−1)),

the space of quadrics in |U| is identified with the space of plane curves of degreed−1. The net of quadricsNis identified with the linear system of first polars of the curveC.

Proof Reversing the proof of property (iii) from Lemma 4.1.3 shows that the image ofC under the mapψ : C → _P(U ⊗V)spans the space. In our case, this implies that the image ofCunder the mapC→ |S2(U∨)|spans the space of quadrics in the dual space. If the image ofCin_P(U)were contained in a quadricQ, then Qwould be apolar to all quadrics in the dual space, a contradiction. Thus the restriction mapris injective. Since the spaces have the same dimension, it must be surjective.

The composition of the mapi:P2→ |O|U|(2)|, x7→Qx,and the isomor- phism|O|U|(2)| ∼=|OP2(d−1)|is a maps :P

2 _{→ |O}

P2(d−1)|. A similar maps0is given by the first polarsx7→Px(C). We have to show that the two maps coincide. Recall thatPx(C)∩C ={c ∈C : x∈Tc(C)}. In the next Lemma we will show that the quadricsQx, x ∈ Tc(C), form the line inN

of quadrics passing through the singular point ofQcequal tor(c). This shows that the quadricQr(x)cuts out inl(C)the divisorr(Px(C)∩C). Thus the curvess(x)ands0(x)of degreed−1cut out the same divisor onC, hence they coincide.

Lemma 4.1.8 Let W ⊂ Sd(U∨) be a linear subspace, and|W|s _{be the}

locus of singular hypersurfaces. Assumex ∈ |W|s_{is a nonsingular point of}

|W|s_{. Then the corresponding hypersurface has a unique ordinary double point} y and the embedded tangent spaceTx(|W|s)is equal to the hyperplane of

hypersurfaces containingy.

Proof AssumeW = Sd_(V∨_{). Then|W|}s _{coincides with the discriminant}

hypersurfaceDd(|U|)of singular degreedhypersurfaces in|U|. If|W| is a proper subspace, then|W|s_{=|W| ∩Dd(|U|). Sincex∈ |W|}s_{is a nonsingular}

point and the intersection is transversal,Tx(|W|s) =Tx(Dd(|U|))∩ |W|. This proves the assertion.

We see that a pair (C, θ), where C is a plane irreducible curve andθ is a non-effective even theta characteristic onC defines a netNof quadrics in |H0(C, θ(1))∨|such thatC = Ns_{. Conversely, letNbe a net of quadrics in}

Pd−1 =|V|. It is known that the singular locus of the discriminant hypersur-

faceD2(d−1)of quadrics inPd−1is of codimension 2. Thus a general net

NintersectsD2(d−1)transversally along a nonsingular curveCof degreed. This gives a representation ofCas a symmetric determinant and hence defines an invertible sheafLand a non-effective even theta characteristicθ. This gives a dominant rational map of varieties of dimension(d2_{+ 3d−16)/2}

G(3, S2(U∨))/PGL(U)− → |O_P2(d)|/PGL(3). (4.8)

The degree of this map is equal to the number of non-effective even theta char- acteristics on a general curve of degree d. We will see in the next chapter that the number of even theta characteristics is equal to2g−1₍₂g_{+ 1), where}

g = (d−1)(d−2)/2is the genus of the curve. A curve C of odd degree

d= 2k+3has a unique vanishing even theta characteristic equal toθ=OC(k) withh0(θ) = (k+ 1)(k+ 2)/2. A general curve of even degree does not have vanishing even theta characteristics.

4.1.4 Contact curves