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Chapter 5: Problem 1

Let \(f\) be a rational function on the surface \(X\). Show that it is possible to "resolve the singularities of \(f^{\prime \prime}\) in the following sense: there is a birational morphism \(g:\) \(X^{\prime} \rightarrow X\) so that \(f\) induces a morphism of \(X^{\prime}\) to \(\mathbf{P}^{1}\). [Hints: Write the divisor of \(f\) as \((f)=\sum n_{i} C_{i} .\) Then apply embedded resolution (3.9) to the curve \(Y=\bigcup C_{i}\) Then blow up further as necessary whenever a curve of zeros meets a curve of poles until the zeros and poles of \(f\) are disjoint.]

Short Answer

Expert verified
By expressing the divisor of \(f\) in sum form, applying the embedded resolution to the curve, and blowing up further as needed, we can resolve the singularities of \(f^{\prime \prime}\). Then, by inducing a birational morphism, we complete the resolution and fulfill the exercise requirements.

Step by step solution

01

Write the divisor

First, we can express the divisor of \(f\) function, denoted as \((f)\), as a sum: \((f)=\sum n_{i} C_{i}\). The divisor of a rational function is the zeros of the function, and in this case it is represented as a sum of \(n_{i}\) sets of \(C_{i}\) where \(C_{i}\) can be any set of points.
02

Apply embedded resolution

Now, apply the process of embedded resolution to the curve \(Y=\bigcup C_{i}\). The embedded resolution is a process of transforming the curve to a simpler form that we can deal with more easily, and in this case, the curve is the union of all \(C_{i}\), denoted as \(\bigcup C_{i}\). This step will help to simplify complex curves into simpler curves.
03

Continue blowing up curves as necessary

Now we continue with the process of blowing up curves whenever a curve of zeros meets a curve of poles. This process is done to ensure that the zeros and poles of function \(f\) are disjoint. This is a geometric technique to simplify an algebraic variety and making the function simpler to deal with.
04

Inducing a morphism

Finally, we're left with creating a birational morphism \(g\) which maps \(X^{\prime}\) to \(X\). As per the definition, a birational morphism is a morphism between two varieties, where its inverse is also a morphism. The 'birational' means that locally (around every point), the map looks like a ratio of polynomials. The problem states that \(X^{\prime} \rightarrow X\) induces a morphism of \(X^{\prime}\) to \(\mathbf{P}^{1}\), which can be interpreted as \(f\) being transformed into \(f_{new}\) in a way that maps \(X^{\prime}\) to \(\mathbf{P}^{1}\).

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Key Concepts

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

Birational Morphism
Understanding birational morphisms lays the foundation for advanced concepts in algebraic geometry. A birational morphism is a powerful tool in the realm of geometrical transformations, as it essentially deals with mappings between two algebraic varieties that are isomorphisms in all but a few points. This means they don't necessarily need to match up perfectly at every single location, much like how a tattered map might still guide you correctly in most places, even if some corners are ripped off.

In the context of our exercise, we create a birational morphism, denoted as, to simplify the function given by eliminating any problematic singularities. By implementing this morphism, we can transform our problematic space into a new, 'cleaned up' one, referred to as , where the function behaves nicely. Imagine having a tricky puzzle - a birational morphism is like reorganizing the pieces to see a clearer picture without changing the original image's content.
Divisor of a Rational Function
The divisor of a rational function gives mathematicians a succinct way to catalog the important parts of a function - namely, where it hits zero and shoots up to infinity. It's like a summary of a long story, boiling down the essence into something more digestible. In symbolism, we represent it as , a linear combination of curve segments, , weighed by . These represent respectively, the places where our function is zero or diverges to infinity.

In the step-by-step solution, this concept is the starting block. Breaking down the function into its zeroes and poles by writing down its divisor, sets the stage for further manipulations. Just as a musician breaks a piece into notes and beats, this step breaks down the function's landscape into manageable parts, each with its own weight or significance, allowing us to later mold it into a form that's easier to harmonize with.
Embedded Resolution
Embedded resolution is akin to untangling a messy knot of string; it systematically simplifies complicated curves until they're manageable. Imagine trying to follow a single thread through a jumbled ball of yarn - that's what dealing with intersecting algebraic curves without embedded resolution is like.

When applied to , we cleanse the individual components of the curve, untying intersections, and ensuring they can be spread out for easier study. This critical step involves a sequence of 'blowing up' operations that methodically pares down the complexity and reveals the underlying structure. It鈥檚 like turning an intricate dance into basic, repeatable steps. By executing this process, we prepare the function for the final transformation into its 'resolved' form.
Blowing Up Curves
The term 'blowing up' in mathematics is more than a vivid metaphor. It's a transformative process where we zoom in on a point or curve to separate overlapping features, much like separating entwined threads by pulling them apart. The blowing up technique is crucial when dealing with curves that intersect, which can obscure the true nature of the function we鈥檙e studying.

In our step-by-step solution, we continue blowing up at every encounter between a zero and a pole to ensure the two never clash. This is essential for resolving the singularities of and involves iterating this process as needed. It鈥檚 the algebraic equivalent of refining a rough gemstone, meticulously cutting away until all facets are distinct and clear, leading us towards a 'smooth' morphism where behaves without complications.

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Most popular questions from this chapter

Let \(X\) be a nonsingular projective variety of any dimension, let \(Y\) be a nonsingular subvariety, and let \(\pi: \widetilde{X} \rightarrow X\) be obtained by blowing up \(Y\). Show that \(p_{a}(\tilde{X})=\) \(p_{a}(X)\)

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 \([5, \mathrm{II}, \mathrm{p} .283]\).

Multiplicity of a Local Ring. (See Nagata \([7, \mathrm{Ch} \text { III, } \S 23]\) or Zariski-Samuel \([1, \text { vol } 2, \mathrm{Ch} \text { VIII, } \S 10] .\) Let \(A\) be a noetherian local ring with maximal ideal \(m.\) For any \(l>0,\) let \(\psi(l)=\operatorname{length}\left(A / m^{2}\right) .\) We call \(\psi\) the Hilbert -Samuel function of \(A\). (a) Show that there is a polynomial \(P_{A}(z) \in \mathbf{Q}[z]\) such that \(P_{A}(l)=\psi(l)\) for all \(l \gg 0 .\) This is the Hilbert-Samuel polynomial of \(A .\) [Hint: Consider the graded \(\left.\operatorname{ring} \operatorname{gr}_{m} A=\oplus_{d \geqslant 0} m^{d} / m^{d+1}, \text { and apply }(1,7.5) .\right]\) (b) Show that \(\operatorname{deg} P_{A}=\operatorname{dim} A\) (c) Let \(n=\operatorname{dim} A .\) Then we define the multiplicity of \(A\), denoted \(\mu(A),\) to be \((n !)\) (leading coefficient of \(P_{A}\) ). If \(P\) is a point on a noetherian scheme \(X\), we define the multiplicity of \(P\) on \(X, \mu_{P}(X),\) to be \(\mu\left(\mathcal{O}_{P, X}\right)\) (d) Show that for a point \(P\) on a curve \(C\) on a surface \(X,\) this definition of \(\mu_{P}(C)\) coincides with the one in the text just before \((3.5 .2)\) (e) If \(Y\) is a variety of degree \(d\) in \(\mathbf{P}^{n}\), show that the vertex of the cone over \(Y\) is a point of multiplicity \(d\)

If \(D\) is an ample divisor on the surface \(X,\) and \(D^{\prime} \equiv D,\) then \(D^{\prime}\) is also ample. Give an example to show, however, that if \(D\) is very ample, \(D^{\prime}\) need not be very ample.

Let \(Y \cong \mathbf{P}^{1}\) be a curve in a surface \(X,\) with \(Y^{2}<0 .\) Show that \(Y\) is contractible \((5.7 .2)\) to a point on a projective variety \(X_{0}\) (in general singular).
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