The Euler sequence is a fundamental concept in algebraic geometry that provides a powerful tool for understanding the geometry of projective spaces. It establishes a connection between the tangent bundle of a projective space and the vector bundle of its homogeneous coordinates. This sequence is particularly useful in studying the cohomology of vector bundles and the construction of important geometric objects like the canonical bundle. In this article, we will delve into the details of the Euler sequence on $\mathbb{P}^n$, exploring its definition, properties, and applications.
The Euler Sequence on $\mathbb{P}^n$
The Euler sequence is a short exact sequence of vector bundles on the projective space $\mathbb{P}^n$, which is defined as follows:
$ 0 \to \mathcal{O}{\mathbb{P}^n}(-1)^{\oplus (n+1)} \to \mathcal{O}{\mathbb{P}^n}^{\oplus (n+1)} \to T_{\mathbb{P}^n} \to 0 $
where:
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$\mathcal{O}_{\mathbb{P}^n}$ denotes the structure sheaf of $\mathbb{P}^n$, which assigns to each open set $U \subset \mathbb{P}^n$ the ring of regular functions on $U$.
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$\mathcal{O}_{\mathbb{P}^n}(-1)$ is the twisted sheaf, which is a line bundle on $\mathbb{P}^n$ defined by:
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For each open set $U \subset \mathbb{P}^n$, the sections of $\mathcal{O}_{\mathbb{P}^n}(-1)$ over $U$ are given by:
$ \mathcal{O}{\mathbb{P}^n}(-1)(U) = { (f_0, f_1, ..., f_n) \in \mathcal{O}{\mathbb{P}^n}(U)^{n+1} | f_i(x) = 0 \text{ for all } x \in U \text{ if } x_i = 0 } $
where $x = [x_0: x_1: ... : x_n]$ denotes the homogeneous coordinates of a point in $\mathbb{P}^n$.
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$T_{\mathbb{P}^n}$ is the tangent bundle of $\mathbb{P}^n$, which assigns to each point $x \in \mathbb{P}^n$ the tangent space $T_x \mathbb{P}^n$.
The maps in the Euler sequence are defined as follows:
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The first map, $\mathcal{O}{\mathbb{P}^n}(-1)^{\oplus (n+1)} \to \mathcal{O}{\mathbb{P}^n}^{\oplus (n+1)}$, is given by the inclusion of sections, where each section $(f_0, f_1, ..., f_n)$ in $\mathcal{O}{\mathbb{P}^n}(-1)^{\oplus (n+1)}$ is mapped to the same section in $\mathcal{O}{\mathbb{P}^n}^{\oplus (n+1)}$.
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The second map, $\mathcal{O}{\mathbb{P}^n}^{\oplus (n+1)} \to T{\mathbb{P}^n}$, is given by the Euler vector field, which maps a section $(f_0, f_1, ..., f_n)$ to the vector field:
$ \sum_{i=0}^n f_i \frac{\partial}{\partial x_i} $
where $\frac{\partial}{\partial x_i}$ denotes the partial derivative with respect to the $i$-th coordinate.
Properties of the Euler Sequence
The Euler sequence has several important properties:
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Exactness: The sequence is exact, meaning that the image of each map is equal to the kernel of the next map. This property ensures that the sequence provides a complete description of the relationship between the involved vector bundles.
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Global Sections: The global sections of the Euler sequence provide information about the geometry of $\mathbb{P}^n$. The global sections of $\mathcal{O}{\mathbb{P}^n}(-1)^{\oplus (n+1)}$ are all zero, while the global sections of $\mathcal{O}{\mathbb{P}^n}^{\oplus (n+1)}$ correspond to the homogeneous coordinates of $\mathbb{P}^n$. The global sections of $T_{\mathbb{P}^n}$ correspond to the vector fields that are tangent to the projective space.
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Twisted Sheaves: The use of the twisted sheaf $\mathcal{O}_{\mathbb{P}^n}(-1)$ in the Euler sequence is crucial for its connection to the geometry of $\mathbb{P}^n$. This sheaf reflects the fact that the tangent space at a point in $\mathbb{P}^n$ is defined by the homogeneous coordinates of that point.
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Dual Sequence: The dual of the Euler sequence gives another important sequence:
$ 0 \to \Omega_{\mathbb{P}^n}^1 \to \mathcal{O}{\mathbb{P}^n}^{\oplus (n+1)} \to \mathcal{O}{\mathbb{P}^n}(1)^{\oplus (n+1)} \to 0 $
where $\Omega_{\mathbb{P}^n}^1$ denotes the cotangent bundle of $\mathbb{P}^n$. This dual sequence is often used to study the differential geometry of projective spaces.
Applications of the Euler Sequence
The Euler sequence has a wide range of applications in algebraic geometry and related fields:
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Cohomology Calculations: The Euler sequence provides a powerful tool for calculating the cohomology groups of vector bundles on $\mathbb{P}^n$. Using the long exact sequence in cohomology associated with the Euler sequence, we can relate the cohomology of the tangent bundle to the cohomology of the twisted sheaves. This is particularly useful for studying the geometry of projective varieties.
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Canonical Bundle: The Euler sequence can be used to compute the canonical bundle of $\mathbb{P}^n$. The canonical bundle, denoted by $\omega_{\mathbb{P}^n}$, is a line bundle that plays a crucial role in the theory of divisors and intersection theory. By taking the determinant of the Euler sequence, we obtain:
$ \omega_{\mathbb{P}^n} = \mathcal{O}_{\mathbb{P}^n}(-n-1) $
This result provides a fundamental link between the geometry of $\mathbb{P}^n$ and its canonical bundle.
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Vector Bundle Constructions: The Euler sequence provides a starting point for constructing various vector bundles on $\mathbb{P}^n$. For example, by taking tensor powers of the twisted sheaves in the Euler sequence, we can obtain other important vector bundles that are closely related to the tangent bundle.
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Intersection Theory: The Euler sequence has implications for intersection theory on $\mathbb{P}^n$. The intersection product of two divisors on $\mathbb{P}^n$ can be expressed in terms of the cohomology classes of the corresponding line bundles. The Euler sequence provides a way to compute these cohomology classes, thereby allowing us to understand the intersection theory on projective spaces.
Conclusion
The Euler sequence on $\mathbb{P}^n$ is a fundamental tool for studying the geometry of projective spaces. Its exactness, global sections, and connections to twisted sheaves provide a powerful framework for understanding the tangent bundle, the canonical bundle, and other important geometric objects. The applications of the Euler sequence extend to cohomology calculations, vector bundle constructions, and intersection theory, highlighting its significance in various areas of algebraic geometry. By understanding the Euler sequence, we gain deeper insights into the structure and properties of projective spaces and the relationships between different geometric objects.
