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Chapter 2: Problem 7

Some Rutional Surfaces. Let \(X=P_{h}^{2},\) and let \(|D|\) be the complete linear system of all divisors of degree 2 on \(X\) (conics). \(D\) corresponds to the invertible sheaf \(\left((2), \text { whose space of global sections has a basis } x^{2}, y^{2}, z^{2}, x y, x z, y=, \text { where } x, y, z\right.\) are the homogeneous coordinates of \(X\) (a) The complete linear system \(|D|\) gives an embedding of \(\mathbf{P}^{2}\) in \(\mathbf{P}^{5}\). whose image is the Veronese surface (I, Ex 2.13 ). (b) Show that the subsystem defined by \(x^{2}, y^{2}, z^{2}, y(x-z),(x-y)=\) gives a closed immersion of \(X\) into \(\mathbf{P}^{4}\). The image is called the Veronese surface in \(\mathbf{P}^{4}\) Cf. \((\mathrm{IV}, \mathrm{Ex} .3 .11)\) (c) Let \(\mathfrak{d} \subseteq|D|\) be the linear system of all conics passing through a fixed point \(P\) Then o gives an immersion of \(L=X-P\) into \(\mathbf{P}^{4}\). Furthermore, if we blow up \(P,\) to get a surface \(\tilde{X}\), then this map extends to give a closed immersion of \(\tilde{X}\) in \(\mathbf{P}^{+}\). Show that \(\tilde{X}\) is a surface of degree 3 in \(\mathbf{P}^{4}\), and that the lines in \(X\) through \(P\) are transformed into straight lines in \(\tilde{X}\) which do not meet. \(\tilde{X}\) is the union of all these lines, so we say \(\tilde{X}\) is a ruled surface (V, 2.19.1).

Short Answer

Expert verified
The complete linear system \(|D|\) corresponds to an embedding of \(P^2\) into \(P^5\) known as the Veronese surface. When the subsystem \(\{x^2, y^2, z^2, y(x-z), (x-y)z\}\) is considered, it creates an immersion of \(X\) into \(P^4\). The point \(P\) is then blown up to create a degree 3 surface \(\tilde{X}\) in \(P^4\), with straight lines through \(P\) being transformed and not meeting in \(\tilde{X}\). This makes \(\tilde{X}\) a ruled surface.

Step by step solution

01

Complete Linear system Embedding

The complete linear system |D|, gives an embedding of \(P^2\) in \(P^5\). This image is the Veronese surface. The basis for the space of global sections of the invertible sheaf ((2) on \(P^2\) are \(x^2, y^2, z^2, xy, xz, yz\).
02

Subsystem defined

The subsystem defined by \(x^2, y^2, z^2, y(x-z),(x-y)z\) gives a closed immersion of \(X\) into \(P^4\). The image here is the Veronese surface in \(P^4\). The basis being chosen significantly impacts the subspace into which it maps this can be seen with the different outcomes in the mappings to \(P^4\) and \(P^5\).
03

Analysis of Linear system through a fixed point

The linear system \(\mathfrak{d}\) of all conics passing through a fixed point \(P\) allows for an immersion of \(X-P\) (subtracting the point \(P\)) into \(P^4\). If one blows up the point \(P\) to get a surface \(\tilde{X}\), this map extends to give a closed immersion of \(\tilde{X}\) in \(P^4\). \(\tilde{X}\) is a surface of degree 3 in \(P^4\). This step involves using the linear system of conics passing through a fixed point, immersing into \(P^4\), and 'blowing up' a point to create a new surface.
04

Conclusion: Properties of the new surface

On the new surface, the lines in \(X\) through point \(P\) are transformed into straight lines in \(\tilde{X}\) which do not meet. \(\tilde{X}\) is the union of all these lines. Therefore, \(\tilde{X}\) is then considered a ruled surface.

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Key Concepts

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

Veronese Surface
The Veronese surface is a fascinating concept in algebraic geometry. It is created by embedding the projective plane \( \mathbf{P}^2 \) into higher-dimensional projective space, specifically \( \mathbf{P}^5 \). This embedding is achieved through the complete linear system \(|D|\) of divisors of degree 2, known as conics, on \( \mathbf{P}^2 \). A linear system provides a family of algebraic varieties, and in this case, it consists of all combinations of the monomials \( x^2, y^2, z^2, xy, xz, yz \). These are the basic functions that form the global sections for this linear system.
The Veronese surface itself represents the image of this embedding. It is significant because it transforms the geometry of the plane into a more intricate shape in higher dimensions. Understanding this transformation helps in researching the geometric properties and intersections of more complex surfaces. It shows how simple quadratic forms can map into rich geometrical structures with more dimensions.
Linear Systems
A linear system in algebraic geometry is essentially a collection of linear combinations of functions that satisfy certain equations. In this context, it usually refers to the set of divisors or the set of hypersurfaces on a variety. For \( \mathbf{P}^2 \), the complete linear system of all degree 2 divisors, known as conics, consists of combinations like \( x^2, y^2, z^2, xy, xz, yz \). This system can create various interesting embeddings and immersions into higher-dimensional spaces like \( \mathbf{P}^4 \) or \( \mathbf{P}^5 \).
When constructing a subsystem, one uses a specific set of divisors that leads to a different outcome or mapping into the space. This simple change in basis can significantly affect the geometry and the type of surface defined by the linear system. These transformations extend to subsystems defined by fewer basis elements or those that pass through a fixed point, impacting the geometry in meaningful ways. Linear systems are crucial for understanding the ways complex surfaces interact in projective spaces.
Blow-up in Algebraic Geometry
The concept of "blow-up" in algebraic geometry is a technique used to resolve singularities and study the structure around points on a variety more closely. In the context of a linear system, consider a set of conics passing through a fixed point \( P \). Removing \( P \) allows for an immersion of the remaining surface, denoted \( X-P \), into projective space \( \mathbf{P}^4 \).
Blowing up \( P \) means replacing it with a new geometric object鈥攁 different variety, \( \tilde{X} \), like constructing a new surface of degree 3 in \( \mathbf{P}^4 \). This process helps smooth out the surface by re-evaluating the nearby geometry. The resulting blown-up surface transforms lines through \( P \) in \( X \) into straight and parallel lines that do not meet in \( \tilde{X} \).
By blowing up, we gain insights into how singularity points influence the properties and intersections of divisors on the surface. It's a valuable method in algebraic geometry because it uncovers the depth of such intersections and the surface's underlying structure, turning complex surface geometry into more accessible parallel constructs.

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Most popular questions from this chapter

V corieties in Projective Space. Let \(h\) be an algebraically closed field, and let \(X\) be a closed subvariety of \(\mathbf{P}_{k}^{n}\) which is nonsingular in codimension one (hence satisfies \((*)\) ). For any divisor \(D=\sum n_{1} Y_{i}\) on \(X\), we define the degree of \(D\) to be \(\sum n_{1}\) deg \(Y_{i},\) where deg \(Y_{i}\) is the degree of \(Y_{1},\) considered as a projective variety itself (I, \(\$ 7\) ) (a) Let \(V\) be an irreducible hypersurface in \(\mathbf{P}^{n}\) which does not contain \(X,\) and let \(Y_{i}\) be the irreducible components of \(V \cap X\). They all have codimension 1 by (I, Ex. 1.8 ). For each \(i\), let \(f_{1}\) be a local equation for \(V\) on some open set \(U\), of \(\mathbf{P}^{n}\) for which \(Y_{i} \cap U_{1} \neq \varnothing,\) and let \(n_{1}=c_{Y},\left(\bar{f}_{1}\right),\) where \(\bar{f}_{1}\) is the restriction of \(f_{i}\) to \(U_{i} \cap X .\) Then we define the dicisor \(V . X\) to be \(\sum n_{i} Y_{i} .\) Extend by linearity, and show that this gives a well-defined homomorphism from the subgroup of Div \(\mathbf{P}^{n}\) consisting of divisors, none of whose components contain \(X,\) to Div \(X\) (b) If \(D\) is a principal divisor on \(\mathbf{P}^{\prime \prime}\), for which \(D . X\) is defined as in (a). show that \(D . X\) is principal on \(X\). Thus we get a homomorphism \(\mathrm{Cl} \mathbf{P}^{n} \rightarrow \mathrm{Cl} X\) (c) Show that the integer \(n_{i}\) defined in (a) is the same as the intersection multiplicity \(i\left(X, V ; Y_{t}\right)\) defined in \((\mathrm{I}, \$ 7) .\) Then use the generalized Bezout theorem (I, 7.7) to show that for any divisor \(D\) on \(P^{\prime \prime}\), none of whose components contain \(X\) \\[ \operatorname{deg}(D . X)=(\operatorname{deg} D) \cdot(\operatorname{deg} X) \\] (d) If \(D\) is a principal divisor on \(X\), show that there is a rational function \(f\) on \(\mathbf{P}^{n}\) such that \(D=(f) . X .\) Conclude that deg \(D=0 .\) Thus the degree function defines a homomorphism deg:Cl \(X \rightarrow\) Z. (This gives another proof of (6.10) since any complete nonsingular curve is projective.) Finally, there is a commutative diagram and in particular. we see that the map \(\mathrm{Cl} \mathrm{P}^{n} \rightarrow \mathrm{Cl} X\) is injective.

(a) Let \(\varphi: \overline{\mathscr{H}} \rightarrow \mathscr{S}\) be a morphism of presheaves such that \(\varphi(\mathcal{U}): \mathscr{F}(U) \rightarrow \mathscr{G}(U)\) is injective for each \(U\). Show that the induced \(\operatorname{map} \varphi^{+}: \overline{\mathscr{H}}^{+} \rightarrow \mathscr{S}^{+}\) of associated sheaves is injective. (b) Use part (a) to show that if \(\varphi: \bar{y} \rightarrow \mathscr{G}\) is a morphism of sheaves, then im \(\varphi\) can be naturally identified with a subsheaf of \(\mathscr{G}\). as mentioned in the text.

Let \(X\) be a variety of dimension \(n\) over \(k .\) Let \(\mathscr{E}\) be a locally free sheaf of \(\operatorname{rank}>n\) on \(X,\) and let \(V \subseteq \Gamma(X, \mathscr{E})\) be a vector space of global sections which generate \(\mathscr{E} .\) Then show that there is an element \(s \in V\), such that for each \(x \in X,\) we have \(s_{x} \notin \mathrm{m}_{x} \mathscr{E}_{x} .\) Conclude that there is a morphism \(\mathscr{O}_{x} \rightarrow \mathscr{E}\) giving rise to an exact sequence \\[ 0 \rightarrow \mathscr{O}_{X} \rightarrow \mathscr{E} \rightarrow \mathscr{E}^{\prime} \rightarrow 0 \\] where \(\mathscr{E}^{\prime}\) is also locally free. \([\text {Hint}:\) Use a method similar to the proof of Bertini's theorem \((8.18) .]\)

(a) Let \(f: X \rightarrow Y\) be a morphism of schemes. Show that \(\mathscr{L} \mapsto f^{*} \mathscr{L}\) induces a homomorphism of Picard groups, \(f^{*}:\) Pic \(Y \rightarrow\) Pic \(X\) (b) If \(f\) is a finite morphism of nonsingular curves, show that this homomorphism corresponds to the homomorphism \(f^{*}: \mathrm{Cl} Y \rightarrow \mathrm{Cl} X\) defined in the text, via the isomorphisms of (6.16) (c) If \(X\) is a locally factorial integral closed subscheme of \(\mathbf{P}_{k}^{n},\) and if \(f: X \rightarrow \mathbf{P}^{n}\) is the inclusion map, then \(f^{*}\) on Pic agrees with the homomorphism on divisor class groups defined in (Ex. 6.2 ) via the isomorphisms of (6.16)

The Topological Space of a Product. Recall that in the category of varieties, the Zariski topology on the product of two varieties is not equal to the product topology (I, Ex. 1.4). Now we see that in the category of schemes, the underlying point set of a product of schemes is not even the product set. (a) Let \(k\) be a field, and let \(\mathbf{A}_{k}^{1}=\operatorname{Spec} k[x]\) be the affine line over \(k\). Show that \(\mathbf{A}_{k}^{1} \times_{\mathrm{Spec} k} \mathbf{A}_{k}^{1} \cong \mathbf{A}_{k}^{2},\) and show that the underlying point set of the product is not the product of the underlying point sets of the factors (even if \(k\) is algebraically closed). (b) Let \(k\) be a field, let \(s\) and \(t\) be indeterminates over \(k .\) Then \(\operatorname{Spec} k(s), \operatorname{Spec} k(t)\) and Spec \(k\) are all one- point spaces. Describe the product scheme Spec \(k(s) \times \operatorname{secc}_{k} \operatorname{Spec} k(t)\).
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