Derived Pullback Of Blowup At A Point In Algebraic Geometry

Introduction

In the realm of algebraic geometry, understanding the behavior of sheaves under various operations is crucial. Among these operations, the pullback plays a fundamental role, allowing us to transport sheaves from one space to another. However, in the context of derived categories, the concept of a derived pullback becomes essential, especially when dealing with singular situations or non-flat morphisms. This article delves into the intricate details of the derived pullback in the specific context of blowing up a point on the projective plane, $\mathbb{P}^2$. We aim to provide a comprehensive explanation of how to compute the derived pullback $Lf^*(\mathcal{O}_p)$, where $f:\tilde{\mathbb{P}^2} \to \mathbb{P}^2$ is the blowup at a point $p \in \mathbb{P}^2$, and to elucidate why the derived pullback differs from the ordinary pullback in this scenario. Our discussion will navigate through the relevant concepts of derived functors, blowups, and derived categories, offering a detailed exploration suitable for both newcomers and seasoned researchers in the field.

Blowups and Their Significance

To fully appreciate the derived pullback in the context of blowups, it is imperative to first understand the concept of a blowup itself. In algebraic geometry, a blowup is a fundamental birational transformation that modifies a variety by replacing a subvariety with the projectivization of its normal cone. This process effectively "resolves" singularities or introduces controlled singularities in a way that allows for a more refined analysis of the original variety. The blowup construction is a cornerstone of resolution of singularities, a central theme in algebraic geometry. The specific case of blowing up a point on the projective plane, $\mathbb{P}^2$, is a classic example that serves as a building block for understanding more complex blowup scenarios. Blowing up a point $p$ in $\mathbb{P}^2$ introduces an exceptional divisor, which is isomorphic to $\mathbb{P}^1$, replacing the point $p$ with a projective line. This transformation significantly alters the local geometry around $p$ and provides a valuable testing ground for various sheaf-theoretic constructions, including the derived pullback.

The blowup $f:\tilde{\mathbb{P}^2} \to \mathbb{P}^2$ at a point $p \in \mathbb{P}^2$ is a birational morphism, meaning that it is an isomorphism outside of a closed subset. In this case, the closed subset is the point $p$, and its preimage under $f$ is the exceptional divisor $E$, which is isomorphic to $\mathbb{P}^1$. The morphism $f$ can be thought of as "zooming in" on the point $p$ and replacing it with the space of tangent directions at $p$. This process is crucial in resolving singularities because it separates tangent directions that might have been indistinguishable in the original space. The blowup construction is not merely a technical tool; it has deep geometric significance. It allows us to study the local behavior of varieties in a more controlled manner and provides a way to simplify complex singularities. Furthermore, blowups play a crucial role in the minimal model program, a central program in birational geometry aimed at classifying algebraic varieties. Therefore, understanding the properties of blowups and their associated operations, such as the derived pullback, is essential for advancing our understanding of algebraic geometry.

Derived Functors and Derived Categories

Before diving into the specifics of the derived pullback, it is crucial to grasp the underlying concepts of derived functors and derived categories. In homological algebra, many functors that arise naturally in algebraic geometry, such as the pullback functor, are not exact. This means that they do not necessarily preserve exact sequences, which are fundamental tools for studying the structure of modules and sheaves. To remedy this issue, we introduce the notion of derived functors, which are a family of functors that extend the original functor and capture the information lost due to non-exactness. The derived pullback, denoted as $Lf^*$, is an example of a derived functor.

The derived pullback $Lf^*$ is defined as the left derived functor of the usual pullback functor $f^*$. To construct derived functors, we embed the category of sheaves into a larger category called the derived category. The derived category, denoted as $D(\mathcal{O}_X)$, is constructed from the category of complexes of sheaves on a variety $X$ by formally inverting quasi-isomorphisms (morphisms that induce isomorphisms on cohomology). This process allows us to work with complexes of sheaves as if they were single objects, and it provides a natural framework for defining derived functors. The derived pullback $Lf^*$ is then defined as the pullback of a complex of sheaves, where the complex is replaced by a quasi-isomorphic complex of flat sheaves (or more generally, sheaves that are acyclic for the pullback functor). This construction ensures that the derived pullback captures the full information about the pullback, including the higher Tor functors that measure the failure of flatness.

The significance of working in the derived category lies in its ability to encode subtle homological information that is lost when working with ordinary functors. For instance, the derived pullback $Lf^*$ takes into account the Tor functors, which measure the failure of the pullback functor to preserve exactness. This is particularly important when dealing with non-flat morphisms, such as the blowup morphism. In such cases, the derived pullback can provide a much more accurate picture of the relationship between sheaves on the original variety and sheaves on the blown-up variety. Moreover, derived categories provide a powerful framework for studying derived equivalences, which are isomorphisms in the derived category that do not necessarily arise from isomorphisms in the underlying categories of sheaves. These equivalences play a crucial role in modern algebraic geometry and have deep connections to representation theory and other areas of mathematics. Therefore, understanding derived functors and derived categories is essential for tackling advanced problems in algebraic geometry and related fields.

The Derived Pullback $Lf^*(\mathcal{O}_p)$

Now, let's focus on the central question: What is the derived pullback $Lf^*(\mathcal{O}_p)$, where $f:\tilde{\mathbb{P}^2} \to \mathbb{P}^2$ is the blowup at a point $p \in \mathbb{P}^2$? Here, $\mathcal{O}_p$ denotes the skyscraper sheaf at the point $p$, which is a sheaf supported only at $p$ with stalk isomorphic to the base field $k$. To compute the derived pullback, we need to find a suitable resolution of $\mathcal{O}_p$ and then apply the pullback functor to this resolution. The skyscraper sheaf $\mathcal{O}_p$ can be resolved by the following complex:

0 \to \mathcal{O}_{\mathbb{P}^2}(-1) \xrightarrow{\phi} \mathcal{O}_{\mathbb{P}^2} \to \mathcal{O}_p \to 0,

where $\phi$ is a section vanishing at $p$. This is a locally free resolution of $\mathcal{O}_p$, which is crucial for computing the derived pullback. The derived pullback $Lf^*(\mathcal{O}_p)$ is then obtained by pulling back this complex:

0 \to f^*\mathcal{O}_{\mathbb{P}^2}(-1) \xrightarrow{f^*\phi} f^*\mathcal{O}_{\mathbb{P}^2} \to 0.

Since $f^*\mathcal{O}_{\mathbb{P}^2}(-1) = \mathcal{O}_{\tilde{\mathbb{P}^2}}(-E)$ and $f^*\mathcal{O}_{\mathbb{P}^2} = \mathcal{O}_{\tilde{\mathbb{P}^2}}$, the complex becomes

0 \to \mathcal{O}_{\tilde{\mathbb{P}^2}}(-E) \xrightarrow{f^*\phi} \mathcal{O}_{\tilde{\mathbb{P}^2}} \to 0,

The map $f^*\phi$ vanishes along the exceptional divisor $E$, so the cokernel of this map is the structure sheaf of the exceptional divisor, $\mathcal{O}_E$. Therefore, the derived pullback $Lf^*(\mathcal{O}_p)$ is quasi-isomorphic to the complex

[\mathcal{O}_{\tilde{\mathbb{P}^2}}(-E) \to \mathcal{O}_{\tilde{\mathbb{P}^2}}],

which has cohomology only in degree 0 and 1. The 0th cohomology is zero, and the 1st cohomology is $\mathcal{O}_E$. In the derived category, this complex is equivalent to $\mathcal{O}_E[-1]$, where $[-1]$ denotes a shift in the derived category. This means that the derived pullback $Lf^*(\mathcal{O}_p)$ is a complex concentrated in degree 1, with cohomology $\mathcal{O}_E$.

Why the Derived Pullback Differs from the Ordinary Pullback

Now, let's address the critical question of why the derived pullback $Lf^*(\mathcal{O}_p)$ differs from the ordinary pullback $f^*(\mathcal{O}_p)$. The ordinary pullback $f^*(\mathcal{O}_p)$ is simply the sheaf-theoretic pullback, which is the sheaf associated to the presheaf $U \mapsto \mathcal{O}_p(f(U))$. In this case, the ordinary pullback $f^*(\mathcal{O}_p)$ is a sheaf supported on the exceptional divisor $E$, but it is not the structure sheaf $\mathcal{O}_E$. Instead, it is a non-trivial extension of $\mathcal{O}_E$ by itself.

The key difference arises from the fact that the pullback functor $f^*$ is not left exact. This means that when we apply $f^*$ to the exact sequence

0 \to \mathcal{O}_{\mathbb{P}^2}(-1) \to \mathcal{O}_{\mathbb{P}^2} \to \mathcal{O}_p \to 0,

the resulting sequence

0 \to f^*\mathcal{O}_{\mathbb{P}^2}(-1) \to f^*\mathcal{O}_{\mathbb{P}^2} \to f^*\mathcal{O}_p \to 0

is not necessarily exact. The failure of exactness is captured by the higher Tor functors, which are precisely what the derived pullback takes into account. In this case, the Tor functor $\mathcal{T}or_1(\mathcal{O}_p, \mathcal{O}_X)$ is non-zero, indicating that the ordinary pullback does not capture the full picture.

The derived pullback, on the other hand, is designed to correct this issue by considering a resolution of $\mathcal{O}_p$ and applying the pullback to the entire complex. This process ensures that we capture the full homological information, including the Tor functors. As we saw in the computation above, the derived pullback $Lf^*(\mathcal{O}_p)$ is quasi-isomorphic to $\mathcal{O}_E[-1]$, which reflects the fact that there is a non-trivial extension in the ordinary pullback that is resolved by working in the derived category. In essence, the derived pullback provides a more refined and accurate description of the relationship between sheaves on $\mathbb{P}^2$ and sheaves on $\tilde{\mathbb{P}^2}$, especially when dealing with non-flat morphisms like the blowup.

Implications and Further Explorations

The computation of the derived pullback $Lf^*(\mathcal{O}_p)$ has significant implications for understanding the geometry of blowups and their effect on sheaves. The fact that the derived pullback is concentrated in degree 1 and has cohomology $\mathcal{O}_E$ reflects the fact that the blowup introduces a new irreducible component (the exceptional divisor) and modifies the local geometry around the blown-up point. This result is a fundamental building block for more advanced computations in algebraic geometry, such as the study of derived categories of coherent sheaves on blowups and the analysis of birational transformations.

Furthermore, this example serves as a paradigm for understanding derived pullbacks in more general contexts. When dealing with non-flat morphisms or singular varieties, the derived pullback often provides a more accurate and informative picture than the ordinary pullback. It captures the homological subtleties that arise from the failure of exactness and provides a powerful tool for studying the derived category of coherent sheaves. For instance, similar techniques can be used to compute the derived pullback of other sheaves, such as structure sheaves of subvarieties or ideal sheaves of singularities. These computations often involve finding suitable resolutions and applying the pullback functor to the resolution, taking into account the higher Tor functors.

In addition to its theoretical significance, the derived pullback has practical applications in various areas of algebraic geometry and related fields. It plays a crucial role in the study of derived equivalences, which are isomorphisms in the derived category that do not necessarily arise from isomorphisms in the underlying categories of sheaves. Derived equivalences have deep connections to representation theory, mirror symmetry, and other areas of mathematics. The derived pullback is also used in the construction of semiorthogonal decompositions, which provide a way to decompose the derived category of a variety into simpler pieces. These decompositions are a powerful tool for studying the structure of the derived category and have applications to the classification of algebraic varieties.

Conclusion

In conclusion, the derived pullback $Lf^*(\mathcal{O}_p)$ of the skyscraper sheaf at a point $p$ under the blowup morphism $f:\tilde{\mathbb{P}^2} \to \mathbb{P}^2$ is a fundamental example that illustrates the importance of derived functors in algebraic geometry. The derived pullback, quasi-isomorphic to $\mathcal{O}_E[-1]$, captures the homological information lost by the ordinary pullback due to the non-flatness of the blowup morphism. This computation highlights the significance of working in the derived category when dealing with singular situations or non-flat morphisms. The derived pullback provides a more refined and accurate description of the relationship between sheaves on the original variety and sheaves on the blown-up variety, and it serves as a building block for more advanced computations in algebraic geometry. Understanding the derived pullback and its properties is crucial for tackling advanced problems in algebraic geometry and related fields, and it opens doors to deeper insights into the geometry of algebraic varieties and their birational transformations. The exploration of derived pullbacks and their applications remains an active area of research, promising further advancements in our understanding of the intricate world of algebraic geometry.