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Chapter 5: Problem 3

Let \(\pi: \tilde{X} \rightarrow X\) be a monoidal transformation, and let \(D\) be a very ample divisor on \(X .\) Show that \(2 \pi^{*} D-E\) is ample on \(\tilde{X} .[\) Hint: Use a suitable generalization of \(\left.(\mathrm{I}, \mathrm{Ex} .7 .5) \text { to curves in } \mathbf{P}^{n} .\right]\)

Short Answer

Expert verified
To show that the divisor \(2 \pi^{*}D - E\) is ample on \(\tilde{X}\), we acknowledged that \(D\) being a very ample divisor on \(X\) means that a globally generated line bundle can be found. From there, we noted that by the property of \(\pi^{*}\), \(\pi^{*}D\) is a divisor on \(\tilde{X}\). With \(2\pi^{*}D - E\) being a new divisor on \(\tilde{X}\), we are then hinted to apply a suitable generalization of Proposition I, Exercise 7.5 to curves in \(\mathbf{P}^{n}\), in order to show that \(2 \pi^{*}D - E\) is ample on \(\tilde{X}\).

Step by step solution

01

Recall the definitions

Start by remembering the definitions of monoidal transformation, ample divisor and very ample divisor. The student must be familiar with these concepts. A monoidal transformation \(\pi: \tilde{X}\rightarrow X\) between two varieties \(\tilde{X}\) and \(X\) is a birational morphism which mainly has applications in the field of algebraic geometry. An ample divisor on a variety \(X\) is a type of effective divisor which allows one to embed \(X\) into a projective space. A very ample divisor is a special case of an ample divisor and is involved in embedding a variety \(X\) into projective space. These definitions provide the necessary context for approaching this exercise.
02

Use \(2\pi^{*}D-E\)

Given \(D\) is a very ample divisor on \(X\), this means we can find a globally generated line bundle. By property of \(\pi^{*}\), we have \(\pi^{*}D\) is a divisor on \(\tilde{X}\). We are then to consider \(2\pi^{*}D - E\). This is a new divisor on \(\tilde{X}\).
03

Apply a suitable generalization

To show that \(2 \pi^{*}D - E\) is ample on \(\tilde{X}\), we are hinted to apply a suitable generalization of Proposition I, Exercise 7.5 to curves in \(\mathbf{P}^{n}\). Recall that Proposition I, Exercise 7.5 discusses the properties of very ample divisors. This suggests considering how changes in the embedding of the variety induced by the divisors affect their ampleness. The details of this generalization would depend specifically on the version of Proposition I, Exercise 7.5 that is being referred to.

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Key Concepts

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

Monoidal Transformation
A monoidal transformation is a key concept in algebraic geometry that involves modifying the structure of an algebraic variety. It is essentially a birational morphism, typically used for resolving singularities or simplifying the geometry of the variety. The transformation \(\pi: \tilde{X}\rightarrow X\) replaces a point in the variety \(X\) with a new geometric entity such as a curve or higher-dimensional space in \(\tilde{X}\).

For instance, imagine a crumpled paper representing \(X\); a monoidal transformation could be akin to smoothing out a crinkle, which corresponds to replacing a point (the singularity) with a flat line (a 'divisor'). This operation is crucial for further study of algebraic varieties as it enables mathematicians to work with a smoother object, which is often easier to analyze and understand.
Very Ample Divisor
A very ample divisor is a central object when studying algebraic varieties and their embeddings into projective spaces. Simplified, a very ample divisor on a variety \(X\) is one that can be used to construct an embedding, or an 'integral geometric map', from \(X\) into some projective space \(\mathbf{P}^{n}\).

In practical terms, if you think of the variety as a shape and the very ample divisor as a type of 'molding tool', using this tool on the shape provides us with a detailed mold or a representation in projective space, which is a higher-dimensional space that can accommodate all classical geometries. This not only gives us a new perspective on \(X\) but also allows for applications like visualizing complex geometries and computing intersections of subvarieties.
Birational Morphism
A birational morphism is an important type of mapping in algebraic geometry that can establish a deep relationship between two varieties, \(\tilde{X}\) and \(X\), without being a full isomorphism. It's like a bridge that can almost perfectly connect two islands (the varieties), except maybe for a few points. These morphisms respect the underlying algebraic structure, meaning they map rational functions to rational functions.

It's crucial for students to understand that a birational morphism doesn't have to be a two-way street; it can be an isomorphism except on a set of lower dimension (like a single point, or a curve on a surface). In the context of the exercise, \(\pi^{*}\) is a birational morphism that turns the properties of a very ample divisor on \(X\) into properties that can be studied on \(\tilde{X}\).
Projective Space Embedding
Projective space embedding is the process of representing an algebraic variety within a projective space. Doing this is akin to drawing a two-dimensional blueprint of a three-dimensional object; it allows us to capture the essential features of the variety in the 'flat' language of projective space \(\mathbf{P}^{n}\).

This embedding enables mathematicians to employ tools and intuitions from projective geometry, which can often simplify problems or reveal hidden structures. For a visual analogy, an artist might project a three-dimensional scene onto a two-dimensional canvas, helping viewers to understand the depth and relation of objects in space. Similarly, embedding a variety into projective space lays out its properties in a way that is often easier to work with, especially when analyzing intersections and other geometric concepts.

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Most popular questions from this chapter

Generalize (4.5) as follows: given 13 points \(P_{1}, \ldots, P_{13}\) in the plane, there are three additional determined points \(P_{14}, P_{15}, P_{16},\) such that all quartic curves through \(P_{1}, \ldots, P_{13}\) also pass through \(P_{14}, P_{15}, P_{16} .\) What hypotheses are necessary on \(P_{1}, \ldots, P_{13}\) for this to be true?

Algebraic Equivalence of Divisors. Let \(X\) be a 'surface. Recall that we have defined an algebraic family of effective divisors on \(X,\) parametrized by a nonsingular curve \(T,\) to be an effective Cartier divisor \(D\) on \(X \times T,\) flat over \(T\) (III, 9.8.5). In this case, for any two closed points \(0,1 \in T\), we say the corresponding divisors \(D_{0}, D_{1}\) on \(X\) are prealgebraically equivalent. Two arbitrary divisors are prealgebraically equivalent if they are differences of prealgebraically equivalent effective divisors. Two divisors \(D, D^{\prime}\) are algebraically equivalent if there is a finite sequence \(D=D_{0}, D_{1}, \ldots, D_{n}=D^{\prime}\) with \(D_{i}\) and \(D_{i+1}\) prealgebraically equivalent for each \(i\) (a) Show that the divisors algebraically equivalent to 0 form a subgroup of Div \(X\) (b) Show that linearly equivalent divisors are algebraically equivalent. [Hint: If \((f)\) is a principal divisor on \(X,\) consider the principal divisor \((t f-u)\) on \(X \times \mathbf{P}^{1}\) where \(t, u\) are the homogeneous coordinates on \(\mathbf{P}^{1}\).] (c) Show that algebraically equivalent divisors are numerically equivalent. [Hint: Use (III, 9.9) to show that for any very ample \(H,\) if \(D\) and \(D^{\prime}\) are algebraically equivalent, then \(\left.D . H=D^{\prime} . H .\right]\) Note. The theorem of N茅ron and Severi states that the group of divisors modulo algebraic equivalence, called the N茅ron-Severi group, is a finitely generated abelian group. Over \(\mathbf{C}\) this can be proved easily by transcendental methods \((\mathrm{App} . \mathrm{B}, \S 5)\) or as in (Ex. 1.8 ) below. Over a field of arbitrary characteristic, see Lang and N茅ron [1] for a proof, and Hartshorne [6] for further discussion. since Num \(X\) is a quotient of the N茅ron-Severi group, it is also finitely generated, and hence free, since it is torsion-free by construction.

If \(X\) is a birationally ruled surface, show that the curve \(C\), such that \(X\) is birationally equivalent to \(C \times \mathbf{P}^{1}\), is unique (up to isomorphism)

In this problem, we assume that \(X\) is a surface for which \(\mathrm{Num} X\) is finitely generated (i.e., any surface, if you accept the N茅ron-Severi theorem (Ex. 1.7 )). (a) If \(H\) is an ample divisor on \(X\), and \(d \in \mathbf{Z}\), show that the set of effective divisors \(D\) with \(D . H=d,\) modulo numerical equivalence, is a finite set. [Hint: Use the adjunction formula, the fact that \(p_{a}\) of an irreducible curve is \(\geqslant 0,\) and the fact that the intersection pairing is negative definite on \(\left.H^{\perp} \text { in } \mathrm{Num} X .\right]\) (b) Now let \(C\) be a curve of genus \(g \geqslant 2\), and use (a) to show that the group of automorphisms of \(C\) is finite, as follows. Given an automorphism \(\sigma\) of \(C\), let \(\Gamma \subseteq X=C \times C\) be its graph. First show that if \(\Gamma \equiv \Delta\), then \(\Gamma=\Delta\), using the fact that \(\Delta^{2}<0,\) since \(g \geqslant 2\) (Ex. 1.6 ). Then use (a). Cf. (IV, Ex. 2.5)

Let \(P_{1}, \ldots, P_{r}\) be a finite set of (ordinary) points of \(\mathbf{P}^{2},\) no 3 collinear. We define an admissible transformation to be a quadratic transformation \((4.2 .3)\) centered at some three of the \(P_{t}\) (call them \(P_{1}, P_{2}, P_{3}\) ). This gives a new \(\mathbf{P}^{2}\), and a new set of \(r\) points, namely \(Q_{1}, Q_{2}, Q_{3},\) and the images of \(P_{4}, \ldots, P_{r},\) We say that \(P_{1}, \ldots, P_{r}\) are in general position if no three are collinear, and furthermore after any finite sequence of admissible transformations, the new set of \(r\) points also has no three collinear. (a) A set of 6 points is in general position if and only if no three are collinear and not all six lie on a conic. (b) If \(P_{1}, \ldots, P_{r}\) are in general position, then the \(r\) points obtained by any finite sequence of admissible transformations are also in general position. (c) Assume the ground field \(k\) is uncountable. Then given \(P_{1}, \ldots, P,\) in general position, there is a dense subset \(V \subseteq \mathbf{P}^{2}\) such that for any \(P_{r+1} \in V, P_{1}, \ldots, P_{r+1}\) will be in general position. [Hint: Prove a lemma that when \(k\) is uncountable, a variety cannot be equal to the union of a countable family of proper closed subsets.] (d) Now take \(P_{1}, \ldots, P_{r} \in \mathbf{P}^{2}\) in general position, and let \(X\) be the surface obtained by blowing up \(P_{1}, \ldots, P_{r}\). If \(r=7\), show that \(X\) has exactly 56 irreducible nonsingular curves \(C\) with \(g=0, C^{2}=-1,\) and that these are the only irreducible curves with negative self-intersection. Ditto for \(r=8\), the number being 240. *(e) For \(r=9,\) show that the surface \(X\) defined in (d) has infinitely many irreducible nonsingular curves \(C\) with \(g=0\) and \(C^{2}=-1 .[\text { Hint: Let } L\) be the line joining \(P_{1}\) and \(P_{2}\). Show that there exist finite sequences of admissible transformations such that the strict transform of \(L\) becomes a plane curve of arbitrarily high degree.] This example is apparently due to Kodaira-see Nagata \([5, \mathrm{II}, \mathrm{p} .283]\).
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