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Chapter 5: Problem 2

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).

Short Answer

Expert verified
Yes, the given curve \(Y\) is indeed contractible to a point on a projective variety \(X_0\) (possibly singular), while lying on surface \(X\), as proven by setting up a homotopy that provides continuous transformations from \(Y\) to \(X_0\).

Step by step solution

01

Understanding the conditions

Firstly, let's clarify the conditions. The curve \(Y\) is isomorphic to \(\mathbf{P}^{1}\) and lies on a surface \(X\). Moreover, a constraint is given by \(Y^{2}<0\).
02

Contractibility of Y

A space is contractible if it is homotopy equivalent to a point. Now impose a contraction \(f: Y \to X_0\), where \(X_0\) is a point in \(X\). Assume that this contraction is continuous and \(f: Y \to X_0\) is a homotopy.
03

Demonstrating Y as contractible

The goal is to show that \(Y\) can be 'shrunk' to a point, i.e., \(X_0\). By properties of homotopy, if Y contracts to a point \(X_0\), there exists a series of continuous transformations taking \(Y\) to \(X_0\) without tearing or gluing, through the homotopy \(f\). The homotopy single out a path in \(X\) from each point in \(Y\) to \(X_0\). Each path should retain continuity and not take any point in \(Y\) out of \(X\). This will satisfy the condition for Y being contractible to \(X_0\) on \(X\). Note that \(X_0\) can be a singular point in a general case.
04

Concluding the solution

Finally, we've shown that curve \(Y\) is contractible to a point \(X_0\) on a projective variety in a continuous manner, while honoring the given conditions about \(Y\) and \(X\). This fulfills the condition of contractibility.

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Key Concepts

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

Projective Variety
In algebraic geometry, a projective variety is an essential concept that represents a closed subset of a projective space. Let鈥檚 break it down! Projective spaces are constructed as an extension of linear spaces that allow better handling of geometric properties at infinity.

A projective variety on such a space typically means a solution set of homogeneous polynomial equations. This set corresponds to points satisfying these equations under projective transformations, ensuring that our analysis accounts for all possible configurations, including those "at infinity." This is often important in curvilinear analysis where your objects might not resemble typical Euclidean structures.
Homotopy
Homotopy is like a bridge between shapes in topology. It shows us how one shape can be continuously transformed into another, without cutting or gluing. Imagine taking a clay model of a dolphin and slowly morphing it into a cat. If you can do this without any breaks, we call these shapes homotopy equivalent. A homotopy between two continuous functions, say from space Y to space X, is a continuous deformation of one function into the other. In our exercise, the homotopy function gives us a way to shrink the space Y to a single point on X. This transformation maps each point in Y continuously to the point on X, preserving the pathways in the process.
Contractibility
Contractibility essentially investigates whether a space can be shrunk to a point within itself. When a space is contractible, like a rubber band, you can compress it down to a single dot without breaking it.

Think of contractibility as "collapsing" a space. If Y is contractible within the surface X to a point X_0, then there鈥檚 a homotopy equivalence between Y and the point X_0, meaning Y can be smoothly compressed down to X_0. The condition of Y虏 < 0 in the problem tells us that, geometrically, it's in a position or arrangement that favors this contraction.

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Most popular questions from this chapter

Let \(H\) be a very ample divisor on the surface \(X,\) corresponding to a projective embedding \(X \subseteq \mathbf{P}^{N} .\) If we write the Hilbert polynomial of \(X\) (III, Ex. 5.2) as \\[ F(z)=\frac{1}{2} a z^{2}+b z+c \\] show that \(a=H^{2}, b=\frac{1}{2} H^{2}+1-\pi,\) where \(\pi\) is the genus of a nonsingular curve representing \(H,\) and \(c=1+p_{a} .\) Thus the degree of \(X\) in \(\mathbf{P}^{N},\) as defined in \((\mathrm{I}, \S 7)\) is just \(H^{2} .\) Show also that if \(C\) is any curve in \(X\), then the degree of \(C\) in \(\mathbf{P}^{N}\) is just \(C . H\)

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 \(d\)

Let \(C, D\) be any two divisors on a surface \(X\), and let the corresponding invertible sheaves be \(\mathscr{L}, \mathscr{M} .\) Show that $$C . D=\chi\left(\mathcal{O}_{X}\right)-\chi\left(\mathscr{L}^{-1}\right)-\chi\left(\mathscr{M}^{-1}\right)+\chi\left(\mathscr{L}^{-1} \otimes \mathscr{M}^{-1}\right)$$

For each of the following singularities at (0,0) in the plane, give an embedded resolution, compute \(\delta_{P},\) and decide which ones are equivalent. (a) \(x^{3}+y^{5}=0\) (b) \(x^{3}+x^{4}+y^{5}=0\) (c) \(x^{3}+y^{4}+y^{5}=0\) (d) \(x^{3}+y^{5}+y^{6}=0\) (e) \(x^{3}+x y^{3}+y^{5}=0\)

Recall that the arithmetic genus of a projective scheme \(D\) of dimension 1 is defined as \(p_{a}=1-\chi\left(\mathcal{O}_{D}\right)(\mathrm{III}, \mathrm{Ex} .5 .3)\) (a) If \(D\) is an effective divisor on the surface \(X\), use (1.6) to show that \(2 p_{a}-2=\) \(D \cdot(D+K)\) (b) \(p_{a}(D)\) depends only on the linear equivalence class of \(D\) on \(X\) (c) More generally, for any divisor \(D\) on \(X\), we define the virtual arithmetic genus (which is equal to the ordinary arithmetic genus if \(D\) is effective) by the same formula: \(2 p_{a}-2=D \cdot(D+K) .\) Show that for any two divisors \(C, D\) we have \\[ p_{a}(-D)=D^{2}-p_{a}(D)+2 \\] and \\[ p_{a}(C+D)=p_{a}(C)+p_{a}(D)+C . D-1 \\]
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