Problem 7 - Algebraic Geometry

Question

Let \(Y\) be an irreducible curve on a surface \(X,\) and suppose there is a morphism \(f: X \rightarrow X_{0}\) to a projective variety \(X_{0}\) of dimension \(2,\) such that \(f(Y)\) is a point \(P\) and \(f^{-1}(P)=Y .\) Then show that \(Y^{2}<0 .[\text { Hint: Let }|H|\) be a very ample (Cartier) divisor class on \(X_{0},\) let \(H_{0} \in|H|\) be a divisor containing \(P,\) and let \(H_{1} \in|H|\) be a divisor not containing \(\left.P . \text { Then consider } f^{*} H_{0}, f^{*} H_{1} \text { and } \hat{H}_{0}=f^{*}\left(H_{0}-P\right)^{-} .\right]\)

Step-by-Step Solution

Answer

Under the given circumstances, the self-intersection number of the irreducible curve \(Y\) is indeed negative.

Step 1: Understanding the given

We have been given an irreducible curve \(Y\) on a surface \(X\), and a morphism \(f: X \rightarrow X_{0}\) to a projective variety \(X_{0}\). Also, we are given divisors \(H_{0}\) and \(H_{1}\) that contain and do not contain \(P\) respectively. Let's also define \(\hat{H}_{0}=f^{*}\left(H_{0}-P\right)^{-}\). We have to prove that \(Y^{2}<0 \).

Step 2: Properties of divisors

First, note that because \(H_{0}\) and \(H_{1}\) are very ample divisors, and since \(H_{0}\) contains \(P\) but \(H_{1}\) does not, we have \(H_{0}^2 > 0\) and \(H_{1}^2 > 0\). Also, \(H_{0} \cdot H_{1} \geq 0\). It's also important to recognize that the pullback \(f^{*}H_{i}\) of the very ample divisors on \(X_{0}\) are likewise very ample divisors on \(X\).

Step 3: Interactions of \(Y\) with \(H_0\) and \(H_1\)

Note that \(Y\) intersects both \(f^{*}H_{0}\) and \(f^{*}H_{1}\). More formally, \(Y \cdot f^{*}H_{i}= d > 0\) for \(i = 0, 1\), since \(f(Y) = P\) is a point and \(P\) is contained by \(H_{0}\) but not \(H_{1}\).

Step 4: Calculating interactions on \(X\)

Now consider \(\hat{H}_{0} = f^{*}(H_{0} - P)^{-}\). Since \(f^{-1}(P) = Y\), we know that \(Y\) cannot intersect \(\hat{H}_{0}\), so \(Y \cdot \hat{H}_{0} \leq 0\).

Step 5: Final computations

If we calculate the self-interaction of \(Y\) on \(X\) using \(\hat{H}_{0}\) and \(f^{*}H_{1}\), we get \(Y^2 = Y \cdot (\hat{H}_{0} + df^{*}H_{1}) \leq Y \cdot \hat{H}_{0} + dY \cdot f^{*}H_{1} \leq 0\). However, since \(Y \cdot f^{*}H_{0} = Y \cdot f^{*}H_{1} = d > 0\), we must have \(Y^2 < 0\).

Step 6: Conclusion

And hence, we have shown that in the given situation, the self-intersection number of an irreducible curve \(Y\) is negative.

Key Concepts

Irreducible Curves

Irreducible curves are a fundamental concept in algebraic geometry, primarily dealing with the properties and behaviors of curves in a given surface. A curve is irreducible if it cannot be expressed as a union of two nontrivial curves; in other words, it is 'whole' and 'complete.' When applied to a surface, these curves denote continuous, non-segmented shapes that cannot be split into simpler parts.

Understanding irreducible curves is crucial when analyzing their interactions with surfaces, especially their intersection properties. When a curve is irreducible on a surface, it typically has certain numerical properties that qualify its intersection behavior with other divisors or curves. For instance, in our example, the exercise explores how the self-intersection of an irreducible curve behaves on a projective surface. This information is used to derive numerical properties like whether the self-intersection number, denoted as \(Y^2\), is negative.

Therefore, mastering the concept of irreducible curves allows students to delve deeper into the interactions of these curves within more complex structures like surfaces and varieties.

Projective Varieties

Projective varieties are one of the pillars of algebraic geometry, providing a way to study solutions to polynomial equations that remain consistent even under perspective transformations. These varieties are essentially a more generalized space that can encapsulate complex shapes and their inherent symmetries. By considering points at infinity, these varieties offer a complete view of geometric objects, addressing discontinuities and undefined behaviors head-on.

In projective geometry, a variety is not merely a set of solutions, but a formal object where geometric intuition coincides with algebraic rigor. For example, in our exercise, the morphism to a projective variety \(X_0\) of dimension \(2\) involves understanding how a surface in more intuitive three-dimensional space can translate, through algebraic methods, into such a shared space. This shift portrays the surface's full spectrum, capturing interactions that might otherwise be overlooked in ordinary spatial dimensions.

• Projective varieties aid in resolving and understanding complex intersections.
• They provide a unified framework to manage and evaluate higher-dimensional geometric entities.

Understanding these core aspects simplifies the transition from algebraic equations to solid geometry, making it indispensable in exploring topics like morphisms and divisors.

Divisors

Divisors play a critical role in algebraic geometry, acting as the building blocks to describe trajectories such as lines, curves, and higher-dimensional cousins in varied surfaces. These are formal sums of codimension-one subvarieties, akin to how numbers are built from their prime factors. Here, they help determine and measure intersections and commonalities within a given surface.

In the context of the exercise, divisors \(H_0\) and \(H_1\) operate as crucial components. They give insight into how intersections occur, particularly when one contains a specified point and the other does not. The properties derived from very ample divisors, which are inherently positive in nature, ensure that they act as lenses to view and understand the internal structure of projective varieties and surfaces.

• Divisors provide a mechanism to track and evaluate intersections within varieties.
• They are essential in determining the numerical properties like positivity and ampleness critical to geometric proof explorations.

By exploring how divisors interact within and across varieties and surfaces, students can better grasp how geometric structures are organized and analyzed in a coherent, methodical way.

Morphisms

Morphisms are mappings or transformations that help relate different algebraic structures, offering a bridge between seemingly unrelated geometric entities. In the exercise, a morphism \(f: X \rightarrow X_0\) provides a connection between the surface \(X\) and its image on a projective variety \(X_0\). It's akin to projecting a three-dimensional object onto a two-dimensional plane but with precise algebraic handles.

Understanding morphisms is about recognizing how to carry one shape into another while preserving intrinsic geometric properties. These functions help maintain relationships, such as point alignment or intersection types, across different geometric settings. For instance, when the morphism \(f\) maps \(Y\) to a point \(P\) on \(X_0\), it effectively condenses the curve into a simpler form without losing essential data.

• Morphisms facilitate the study of surfaces and varieties by providing a consistent method of mapping between spaces.
• They help in simplifying complex interactions and translating them into more manageable forms.

By examining how morphisms function and maintain essential properties, students can better appreciate the multi-layered relationships within algebraic geometry. This understanding is key in exploring how different geometric and algebraic constructs interact and coincide across mathematical contexts.