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Chapter 4: Problem 5

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].

Short Answer

Expert verified
The genus 5 curves in a canonical model give rise to a 12-dimensional family. Genus 5 curves with a \(g_{3}^{1}\) form an 11-dimensional irreducible family mapped onto a plane quintic with a node. In this case, the canonical curve is part of a cubic surface \(V\) that contains all the trisecants of the curve and resides within quadric hypersurfaces covering the curve. If \(g_{3}^{1}\) is absent, the canonical model is a complete intersection. Conforming to the Enriques-Petri theorem, for any nonhyperelliptic curve of genus \(g \geqslant 3,\) the canonical model is projectively normal and intersects quadric hypersurfaces unless it harbors a \(g_{3}^{1}\) or stands at \(g=6\) while containing a \(g_{5}^{2}\).

Step by step solution

01

Curves of Genus 5 as Complete Intersections

The exercise starts with setting up the situation where \(X\) is a curve of genus 5 which is not hyperelliptic. Its canonical model in \(\mathbf{P}^{4}\) is a complete intersection \(F_{2} \cdot F_{2} \cdot F_{2}\). This configuration forms a family of dimension 12.
02

Presence of \(g_{3}^{1}\)

The curve \(X\) has a \(g_{3}^{1}\) if and only if it can be considered as a plane quintic with one node. As the hint suggests, if \(D \in g_{3}^{1}\), apply \(K-D\) to map \(X \rightarrow \mathbf{P}^{2}\). The curves satisfying this condition form an irreducible family of dimension 11.
03

Canonical System through Conics

In the scenario where \(X\) has \(g_{3}^{1}\), the conics passing through the node define the canonical system, disregarding the fixed points at the node. By utilizing the linear system of these conics, \(X\) is shown to be included in a cubic surface \(V \subseteq \mathbf{P}^{4}\), with \(V\) identical to \(\mathbf{P}^{2}\) with an added point.
04

Consideration of Trisecants and Uniqueness

The cubic surface \(V\) is also described to be the collection of all the trisecants of \(X\) connected to the \(g_{3}^{1}\), rendering \(V\) and the \(g_{3}^{1}\) to be unique and confined to the intersection of all the quadric hypersurfaces encapsulating \(X\).
05

Scenario with Absence of \(g_{3}^{1}\) and Theorem of Enriques and Petri

If \(X\) does not have a \(g_{3}^{1}\), then the canonical embedding of \(X\) is designated to be a complete intersection, as described in part (a). The theorem by Enriques and Petri states that for a non-hyperelliptic curve of genus \(g \geqslant 3,\), the canonical model is projectively normal. It is an intersection of quadric hypersurfaces unless \(X\) has a \(g_{3}^{1}\) or \(g=6\) and \(X\) has a \(g_{5}^{2}\).

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Key Concepts

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

Genus of a Curve
In algebraic geometry, the concept of the genus of a curve plays a vital role. It essentially measures the curve's complexity, akin to the number of holes. For instance, a sphere has genus 0, while a torus has genus 1. A curve of genus 5, as mentioned in the problem, means it has a complexity comparable to having "5 holes".

Understanding the genus helps us classify and study curves based on their attributes. Curves of different genera behave differently in geometric contexts, affecting properties like symmetry and embedding in projective spaces. The genus is determined using the Riemann-Roch theorem in complex curves, helping mathematicians explore various configurations and relations within the curve space.
Non-Hyperelliptic Curves
Non-hyperelliptic curves are a distinct category within algebraic geometry. A hyperelliptic curve can be mapped onto a line or a conic in a way that identifies pairs of points. Curves that do not fit this definition are non-hyperelliptic. In genus 5, non-hyperelliptic curves are central to the study.

These curves cannot be expressed as a double cover of a conic. Consequently, they possess more complex configurations and require a different approach to their study in projective geometry. In particular, their canonical models diverge from hyperelliptic ones, leading to unique properties and dimensional configurations. These distinctive characteristics are pivotal in understanding the non-hyperelliptic curve behavior in canonical embeddings.
Canonical Models
Canonical models of a curve are embeddings of the curve into a projective space, giving it a shape or "model." For a curve of genus 5 which is non-hyperelliptic, the canonical model lives in \( \mathbf{P}^{4} \).

Using the complete linear system associated with the canonical divisor, a curve can be embedded in such a way that essential properties become apparent and manageable. This embedding offers a clearer perspective on the geometric and arithmetic properties of the curve.

The canonical model's power lies in its uniformity and normality, making it a cornerstone in the study of algebraic curves. The embedding decision affects the modifications necessary to study intersections, such as complete intersections explored in genus 5 without any presence of \( g_{3}^{1} \).
Complete Intersections
The term "complete intersection" in algebraic geometry refers to curves or varieties that can be represented as intersections of hypersurfaces. For instance, in our discussion, a genus 5 curve can be built from the intersection of three quadrics \( F_{2} \cdot F_{2} \cdot F_{2} \) in \( \mathbf{P}^{4} \).

Such a configuration is not just interesting mathematically but also offers insights into the dimensionality and properties of families of curves. Complete intersections have an intrinsic ability to meet all significant expectations "without surplus," making them a robust foundation for studying curve families that are non-hyperelliptic in nature.

By exploring these configurations, mathematicians can infer about the family sizes and dimensions, contributing to a better understanding of the algebraic structures that underpin these curves.

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Most popular questions from this chapter

For an effective divisor \(D\) on a curve \(X\) of genus \(g\), show that \(\operatorname{dim}|D| \leqslant \operatorname{deg} D\) Furthermore, equality holds if and only if \(D=0\) or \(g=0\).

In view of \((3.10),\) one might ask conversely, is every plane curve with nodes a projection of a nonsingular curve in \(\mathbf{P}^{3}\) ? Show that the curve \(x y+x^{4}+y^{4}=0\) (assume char \(k \neq 2\) ) gives a counterexample.

Let \(X, P_{0}\) be an elliptic curve having an endomorphism \(f: X \rightarrow X\) of degree 2 (a) If we represent \(X\) as a \(2-1\) covering of \(\mathbf{P}^{1}\) by a morphism \(\pi: X \rightarrow \mathbf{P}^{1}\) ramified at \(P_{0},\) then as in \((4.4),\) show that there is another morphism \(\pi^{\prime}: X \rightarrow \mathbf{P}^{1}\) and a morphism \(g: \mathbf{P}^{1} \rightarrow \mathbf{P}^{1},\) also of degree \(2,\) such that \(\pi \quad f=g \quad \pi^{\prime}\) (b) For suitable choices of coordinates in the two copies of \(\mathbf{P}^{1}\), show that \(y\) can be taken to be the morphism \(x \rightarrow x^{2}\) (c) Now show that \(g\) is branched over two of the branch points of \(\pi\), and that \(g^{-1}\) of the other two branch points of \(\pi\) consists of the four branch points of \(\pi\). Deduce a relation involving the invariant , of \(X\). (d) Solving the above, show that there are just three values of \(j\) corresponding to elliptic curves with an endomorphism of degree \(2,\) and find the corresponding values of \(\lambda\) and \(j.\)

Let \(X\) be an irreducible nonsingular curve in \(\mathbf{P}\) '. Then for each \(m>>0\), there is a nonsingular surface \(F\) of degree \(m\) containing \(X\). [Hint: Let \(\pi: \tilde{\mathbf{P}} \rightarrow \mathbf{P}^{3}\) be the blowing-up of \(X\) and let \(Y=\pi^{-1}(X) .\) Apply Bertini's theorem to the projective embedding of \(\left.\tilde{\mathbf{P}} \text { corresponding to } \mathscr{I}_{Y} \otimes \pi^{*} C_{p}(m) .\right]\)

Classification of Curres of Genus 2. Fix an algebraically closed field \(k\) of characteristic \(\neq 2\) (a) If \(X\) is a curve of genus 2 over \(k\), the canonical linear system \(|K|\) determines a finite morphism \(f: X \rightarrow \mathbf{P}^{1}\) of degree 2 (Ex. 1.7 ). Show that it is ramified at exactly 6 points, with ramification index 2 at each one. Note that \(f\) is uniquely determined, up to an automorphism of \(\mathbf{P}^{1},\) so \(X\) determines an (unordered) set of 6 points of \(\mathbf{P}^{1},\) up to an automorphism of \(\mathbf{P}^{1}\). (b) Conversely, given six distinct elements \(\alpha_{1}, \ldots, \alpha_{6} \in k,\) let \(K\) be the extension of \(k(x)\) determined by the equation \(z^{2}=\left(x-\alpha_{1}\right) \cdots\left(x-\alpha_{6}\right) .\) Let \(f: X \rightarrow \mathbf{P}^{1}\) be the corresponding morphism of curves. Show that \(g(X)=2,\) the map \(f\) is the same as the one determined by the canonical linear system, and \(f\) is ramified over the six points \(x=\alpha_{i}\) of \(\mathbf{P}^{1}\), and nowhere else. (Cf. (II, Ex. 6.4 ).) (c) Using (I, Ex. 6.6), show that if \(P_{1}, P_{2}, P_{3}\) are three distinct points of \(\mathbf{P}^{1}\), then there exists a unique \(\varphi \in\) Aut \(\mathbf{P}^{1}\) such that \(\varphi\left(P_{1}\right)=0, \varphi\left(P_{2}\right)=1, \varphi\left(P_{3}\right)=x\) Thus in (a), if we order the six points of \(\mathbf{P}^{1}\), and then normalize by sending the first three to \(0,1, x,\) respectively, we may assume that \(X\) is ramified over \(0,1, \nsim, \beta_{1}, \beta_{2}, \beta_{3},\) where \(\beta_{1}, \beta_{2}, \beta_{3}\) are three distinct elements of \(k, \neq 0,1\) (d) Let \(\Sigma_{6}\) be the symmetric group on 6 letters. Define an action of \(\Sigma_{6}\) on sets of three distinct elements \(\beta_{1}, \beta_{2}, \beta_{3}\) of \(k, \neq 0,1,\) as follows: reorder the set \(0,1, x, \beta_{1}, \beta_{2}, \beta_{3}\) according to a given element \(\sigma \in \Sigma_{6},\) then renormalize as in \((\mathrm{c})\) so that the first three become \(0,1, x\) again. Then the last three are the new \(\beta_{1}^{\prime}, \beta_{2}^{\prime}, \beta_{3}^{\prime}\) (e) Summing up, conclude that there is a one-to-one correspondence between the set of isomorphism classes of curves of genus 2 over \(k\), and triples of distinct elements \(\beta_{1}, \beta_{2}, \beta_{3}\) of \(k, \neq 0,1,\) modulo the action of \(\Sigma_{6}\) described in (d). In particular, there are many non-isomorphic curves of genus \(2 .\) We say that curves of genus 2 depend on three parameters, since they correspond to the points of an open subset of \(\mathbf{A}_{k}^{3}\) modulo a finite group
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