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Chapter 4: Problem 2

If \(D\) is any divisor of degree \(\geqslant 3\) on the elliptic curve \(X\), and if we embed \(X\) in \(\mathbf{P}^{n}\) by the complete linear system \(|D|\), show that the image of \(X\) in \(\mathbf{P}^{n}\) is projectively normal. Note. It is true more generally that if \(D\) is a divisor of degree \(\geqslant 2 g+1\) on a curve of genus \(y\), then the embedding of \(X\) by \(|D|\) is projectively normal (Mumford \([4, p .55]\)

Short Answer

Expert verified
By utilizing the general theorem, it's seen that the image of an elliptic curve \(X\) embedded in Projective space \(\mathbf{P}^{n}\) via the complete linear system \(|D|\), where \(D\) is any divisor of degree \(\geqslant 3\), is projectively normal.

Step by step solution

01

Understand Pre-requisites

Firstly, understand that an elliptic curve \(X\) over a field \(k\) is a projective algebraic curve of genus \(1\) together with a specified point at infinity. A divisor \(D\) on this curve is a formal sum of points on \(X\) (with algebraic coefficients), and the degree of a divisor is the sum of its coefficients. A linear system \( |D| \) is the set of effective divisors linearly equivalent to \(D\). When we say that \(X\) is embedded in projective space \(\mathbf{P}^{n}\) via \(|D|\), it means that each point on \(X\) is associated with a point in \(\mathbf{P}^{n}\) based on the elements of \(|D|\).
02

Define projective normality

Projective normality is a property of an algebraic variety embedded in projective space. An embedding \( i: X \rightarrow P^n \) of an algebraic variety \( X \) is said to be projectively normal if the ring of regular functions \( S(X) \) on \( X \) is integrally closed in its field of fractions. In simpler terms, an embedded variety is projectively normal if it can't be defined by fewer homogeneous equations than the dimension of the projective space it is embedded in.
03

Apply projective normality to this problem

Take any divisor \(D\) on \(X\) of degree \(\geqslant 3\). Then the condition implies that the coordinate ring \(S(D)\) of the system \(|D|\), which is the ring of global sections of the sheaf of all divisors linearly equivalent to \(D\), must be integrally closed. Since \(D\) is of degree \(\geqslant 3\) (which is \(\geqslant 2 \cdot 1 + 1\)), the general theorem mentioned in the question (Mumford \([4, p .55]\)) applies, and so the image of \(X\) under the embedding defined by \(|D|\) is indeed projectively normal.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Elliptic Curve
An elliptic curve is a type of smooth, projective algebraic curve that has a distinct set of characteristics. It is defined over a field (often the field of complex numbers), and must satisfy a specific cubic equation. One of the most important features is that it has genus 1, which in the realm of algebraic geometry, means it technically has one 'hole'. Additionally, every elliptic curve has at least one rational point that serves as the 'point at infinity'.

Elliptic curves have numerous applications ranging from number theory to cryptography. They play a pivotal role in Andrew Wiles' proof of Fermat's Last Theorem, and are critical in the construction of elliptic curve cryptography (ECC). Understanding elliptic curves is fundamental to exploring more complex concepts in algebraic geometry and number theory.
Divisor on a Curve
In algebraic geometry, a divisor on a curve serves as an important tool when considering the modular aspects of a curve. It is a formal sum of points on the curve with integer coefficients. On an elliptic curve, the sum of the coefficients defines the divisor's degree. Given a divisor D, one can construct the linear system |D|, which is the collection of all divisors that are equivalent to D under the curve's addition law.

A crucial property of divisors is that they help to classify line bundles, which in turn, control the ways one can map or embed a curve into projective space. For an elliptic curve, which is inherently one-dimensional, divisors play a critical role in understanding the structure and features of the curve.
Linear System
A linear system is a central concept in algebraic geometry as it is closely related to the maps that take an algebraic curve to projective space. Formally, a linear system on a divisor D is the complete set of linearly equivalent divisors, denoted by |D|.

This concept is visualized as the set of hypersurfaces that intersect the curve in a way that corresponds to the divisor. When we embed a curve into projective space using a linear system, we can view each point on the curve being represented by a hyperplane that it specifies. This system informs the manner and the space into which the curve is embedded.
Algebraic Geometry
Finally, algebraic geometry is the mathematical study that elegantly bridges abstract algebra with geometry. It deals specifically with the solutions of systems of algebraic equations and the geometric structures that emerge from them, such as varieties, which include curves, surfaces, and more complex geometric constructs. It intertwines concepts from various branches of mathematics to provide a deeper understanding of geometric objects.

Algebraic geometry provides us with the necessary tools and language to discuss projective normality as it pertains to embeddings of varieties (like our elliptic curve) into projective space. In this light, projective normality asserts a robust algebraically-defined property of these embeddings, ensuring that the geometric object retains its integrity in the higher-dimensional space.

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Most popular questions from this chapter

Curves of Genus \(5 .\) Assume \(X\) is not hyperelliptic. (a) The curves of genus 5 whose canonical model in \(\mathbf{P}^{4}\) is a complete intersection \(F_{2} \cdot F_{2} \cdot F_{2}\) form a family of dimension 12 (b) \(X\) has a \(g_{3}^{1}\) if and only if it can be represented as a plane quintic with one node. These form an irreducible family of dimension 11. [Hint: If \(D \in g_{3}^{1},\) use \(K-D\) to \(\left.\operatorname{map} X \rightarrow \mathbf{P}^{2} .\right]\) "(c) In that case, the conics through the node cut out the canonical system (not counting the fixed points at the node). Mapping \(\mathbf{P}^{2} \rightarrow \mathbf{P}^{4}\) by this linear system of conics, show that the canonical curve \(X\) is contained in a cubic surface \(V \subseteq \mathbf{P}^{4},\) with \(V\) isomorphic to \(\mathbf{P}^{2}\) with one point blown up (II, Ex. 7.7). Furthermore, \(V\) is the union of all the trisecants of \(X\) corresponding to the \(g_{3}^{1}(5.5 .3)\) so \(V\) is contained in the intersection of all the quadric hypersurfaces containing \(X .\) Thus \(V\) and the \(g_{3}^{1}\) are unique. Note. Conversely, if \(X\) does not have a \(g_{3}^{1},\) then its canonical embedding is a complete intersection, as in (a). More generally, a classical theorem of Enriques and Petri shows that for any nonhyperelliptic curve of genus \(g \geqslant 3,\) the canonical model is projectively normal, and it is an intersection of quadric hypersurfaces unless \(X\) has a \(g_{3}^{1}\) or \(g=6\) and \(X\) has a \(g_{5}^{2} .\) See Saint-Donat [1].

Etale Corers of Degree \(2 .\) Let \(Y\) be a curve over a field \(k\) of characteristic \(\neq 2\) We show there is a one-to-one correspondence between finite 茅tale morphisms \(f: X \rightarrow Y\) of degree \(2,\) and 2 -torsion elements of Pic \(Y,\) i.e., invertible sheaves \(\mathscr{L}\) on \(Y\) with \(\mathscr{L}^{2} \cong \mathcal{O}_{\mathbf{Y}}\) (a) Given an 茅tale morphism \(f: X \rightarrow Y\) of degree \(2,\) there is a natural map \(C_{Y} \rightarrow\) \(f_{*} C_{X}\). Let \(\mathscr{L}\) be the cokernel. Then \(\mathscr{L}\) is an invertible sheafon \(Y, \mathscr{L} \cong \operatorname{det} f_{*} \mathscr{C}_{X}\) and so \(\mathscr{L}^{2} \cong \mathscr{C}_{Y}\) by (Ex. 2.6). Thus an 茅tale cover of degree 2 determines a 2-torsion element in Pic \(Y\) (b) Conversely, given a 2 -torsion element \(\mathscr{L}\) in Pic \(Y\), define an \(\mathscr{C}_{Y}\) -algebra structure on \(\mathscr{U}_{Y} \oplus \mathscr{L}\) by \(\langle a, b\rangle \cdot\left\langle a^{\prime}, b^{\prime}\right\rangle=\left\langle a a^{\prime}+\varphi\left(b \otimes b^{\prime}\right), a b^{\prime}+a^{\prime} b\right\rangle,\) where \(\varphi\) is an isomorphism of \(\mathscr{L} \otimes \mathscr{L} \rightarrow \mathscr{O}_{Y} .\) Then take \(X=\operatorname{Spec}\left(\mathcal{C}_{Y} \oplus \mathscr{L}\right)(\mathrm{II}, \mathrm{Ex} .5 .17)\) Show that \(X\) is an 茅tale cover of \(Y\) (c) Show that these two processes are inverse to each other. [Hint: Let \(\tau: X \rightarrow X\) be the involution which interchanges the points of each fibre of \(f\). Use the trace map \(a \mapsto a+\tau(a)\) from \(f_{*} \mathscr{C}_{x} \rightarrow \mathscr{C}_{y}\) to show that the sequence of \(\mathscr{C}_{Y^{-}}\) modules in (a) $$0 \rightarrow \mathscr{O}_{r} \rightarrow f_{*} \mathscr{O}_{X} \rightarrow \mathscr{L} \rightarrow 0$$ is split exact. Note. This is a special case of the more general fact that for \((n, \operatorname{char} k)=1,\) the 茅tale Galois covers of \(Y\) with group \(\mathbf{Z} / n \mathbf{Z}\) are classified by the 茅tale cohomology \(\operatorname{group} H_{\mathrm{er}}^{1}(Y, \mathbf{Z} / n \mathbf{Z}),\) which is equal to the group of \(n\) -torsion points of Pic \(Y .\) See Serre [6]

A Funny Curre in Characteristic \(p\). Let \(X\) be the plane quartic curve \(x^{3} y+y^{3} z+\) \(z^{3} x=0\) over a field of characteristic \(3 .\) Show that \(X\) is nonsingular, every point of \(X\) is an inflection point, the dual curve \(X^{*}\) is isomorphic to \(X,\) but the natural \(\operatorname{map} X \rightarrow X^{*}\) is purely inseparable.

If \(X\) is a curve of genus \(\geqslant 2\) over a field of characteristic \(0,\) show that the group Aut \(X\) of automorphisms of \(X\) is finite. \([\text {Hint}: \text { If } X\) is hyperelliptic, use the unique \(g_{2}^{1}\) and show that Aut \(X\) permutes the ramification points of the 2 -fold covering \(X \rightarrow \mathbf{P}^{1} .\) If \(X\) is not hyperelliptic, show that Aut \(X\) permutes the hyperosculation points (Ex. \(4.6 \text { ) of the canonical embedding. Cf. (Ex. } 2.5 \text { ). }]\)

A rational curve of degree 4 in \(\mathbf{P}^{3}\) is contained in a unique quadric surface \(Q,\) and \(Q\) is necessarily nonsingular.
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