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Chapter 4: Problem 8

A curve \(X\) of genus \(g\) has a nonspecial divisor \(D\) of degree \(d\) such that \(|D|\) has no base points if and only if \(d \geqslant q+1\).

Short Answer

Expert verified
The nonspecial divisor \(D\) of a curve \(X\) of genus \(g\) requires the degree \(d\) to be equal to or greater than \(q+1\) such that the linear system \(\|D\|\) has no base points.

Step by step solution

01

Definition of nonspecial divisor

A nonspecial divisor is defined as a divisor for which \(l(D) = deg(D) + 1 - g\), where \(l(D)\) is the dimension of the space of meromorphic functions with pole order at most that given by \(D\) and 'deg' is the degree of the divisor.
02

Understanding base points

Base points of \(\|D\|\) are defined as the points \(P\) such that every divisor \(D'\) in \(\|D\|\) passes through \(P\). If \(\|D\|\) has no base points, then there is some divisor in the system that does not pass through every point of the curve.
03

Applying the given condition

According to the given condition, \(\|D\|\) has no base points if and only if \(d \geqslant q+1\). Note we have to consider \(q\) as the smallest positive integer for which \(l(D-qP) < l(D)\) for all \(P\) on the curve, i.e., \(q\) is the smallest positive integer such that adding \(q\) times any point \(P\) to \(D\) does not add any new function to \(l(D)\).
04

Conclusion

We can conclude that a nonspecial divisor \(D\) of degree \(d\) on a curve \(X\) of genus \(g\) implies that \(\|D\|\) has no base points if and only if \(d \geqslant q+1\). This condition gives us a criterion for when the linear system \(\|D\|\) is base point free.

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Key Concepts

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

Genus of a Curve
The genus of a curve is a fundamental concept in algebraic geometry. It is essentially a measure of the curve's complexity. For instance, a curve with no holes, like a simple circle or a line, has a genus of 0. In contrast, a torus (imagine the shape of a donut) has a genus of 1.

Genus can be thought of as the number of "holes" or "handles" in a surface:
  • A genus of 0 implies a surface without holes.
  • Each "handle" on the surface corresponds to one increment in the genus.
In algebraic terms, the genus \(g\) is an invariant that does not change when the curve is transformed by smooth deformations. It's intimately connected to several characteristics of a curve, such as the number of independent loops on the surface.
Nonspecial Divisor
In algebraic geometry, divisors are formal sums of points on a curve. A nonspecial divisor is a specific type of divisor that has certain properties. It satisfies the condition \(l(D) = \text{deg}(D) + 1 - g\), where \(l(D)\) is the dimension of the space of meromorphic functions on the curve with poles determined by the divisor \(D\).

Here's a simple breakdown:
  • Meromorphic functions are like rational functions, allowing for poles at specific points determined by the divisor.
  • The degree of a divisor, denoted as \(\text{deg}(D)\), is the sum of coefficient values from this formal sum.
  • The special property of a nonspecial divisor is that it achieves the maximum possible dimension for the space of functions it determines.
Understanding nonspecial divisors helps identify particular characteristics of curves and predict their function possibilities.
Base Points
Base points are critical in understanding the behavior of divisors on a curve. Specifically, a base point is a point on the curve that every divisor within a given linear system must pass through. If \(\|D\|\), the set of all effective divisors linearly equivalent to \(D\), has base points, it requires every divisor in the system to intersect at those points.

Consider these key points:
  • If there are no base points, at least one divisor in \(\|D\|\) does not pass through every point.
  • A system being base point free indicates more freedom in choosing which points divisors can pass through.
  • This concept helps determine when and how the divisors cover the curve.
Knowing whether base points exist is essential for tasks like embedding curves in projective space.
Degree of a Divisor
The degree of a divisor is one of the most straightforward yet essential concepts. It is the sum of the coefficients in a formal sum of points. For instance, if the divisor is \(D = a_1P_1 + a_2P_2 + \ldots + a_nP_n\), then \(\text{deg}(D) = a_1 + a_2 + \ldots + a_n\).

Here鈥檚 why degree matters:
  • It measures the order or "weight" of the divisor along the curve.
  • The degree impacts the function space \(l(D)\), which is essential in the Riemann-Roch theorem, a cornerstone of algebraic geometry.
  • In simpler terms, the degree can determine how divisors "behave" when covering or interacting with the curve.
Thus, understanding the degree is vital for analyzing and predicting the properties of curves in algebraic geometry.

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Most popular questions from this chapter

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 \(j.\)

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].

Curves of Degree 4 (a) If \(X\) is a curve of degree 4 in some \(\mathbf{P}^{n}\), show that either (1) \(g=0,\) in which case \(X\) is either the rational normal quartic in \(\mathbf{P}^{4}\) (Ex. 3.4 ) or the rational quartic curve in \(\mathbf{P}^{3}(\mathrm{II}, 7.8 .6),\) or (2) \(X \subseteq \mathbf{P}^{2},\) in which case \(g=3,\) or (3) \(X \subseteq \mathbf{P}^{3}\) and \(g=1\) (b) In the case \(g=1,\) show that \(X\) is a complete intersection of two irreducible quadric surfaces in \(\mathbf{P}^{3}\) (I, Ex. 5.11). \(\left[\text { Hint: Use the exact sequence } 0 \rightarrow \mathscr{I}_{X} \rightarrow\right.\) \(\mathcal{O}_{\mathbf{P}^{3}} \rightarrow \mathcal{O}_{X} \rightarrow 0\) to compute \(\operatorname{dim} H^{0}\left(\mathbf{P}^{3}, \mathscr{T}_{X}(2)\right),\) and thus conclude that \(X\) is contained in at least two irreducible quadric surfaces.]

We say a (singular) integral curve in \(\mathbf{P}^{n}\) is strange if there is a point which lies on all the tangent lines at nonsingular points of the curve. (a) There are many singular strange curves, e.g., the curve given parametrically by \(x=t, y=t^{p}, z=t^{2 p}\) over a field of characteristic \(p>0\). (b) Show, however, that if char \(k=0,\) there aren't even any singular strange curves besides \(\mathbf{P}^{1}\).

Let \(X\) be a curve of genus \(g .\) Show that there is a finite morphism \(f: X \rightarrow \mathbf{P}^{1}\) of degree \(\leqslant g+1 .\) (Recall that the degree of a finite morphism of curves \(f: X \rightarrow Y\) is defined as the degree of the field extension \([K(X): K(Y)](\mathrm{II}, \S 6) .)\)
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