The Tautological Bundle On Projective Space
One of the most important line bundles in algebraic geometry is the tautological line bundle on projective space. The projectivization P(V) of a vector space V over a field k is defined to be the quotient of by the action of the multiplicative group k×. Each point of P(V) therefore corresponds to a copy of k×, and these copies of k× can be assembled into a k×-bundle over P(V). k× differs from k only by a single point, and by adjoining that point to each fiber, we get a line bundle on P(V). This line bundle is called the tautological line bundle. This line bundle is sometimes denoted since it corresponds to the dual of the Serre twisting sheaf .
Read more about this topic: Line Bundle
Famous quotes containing the words bundle and/or space:
“In the quilts I had found good objects—hospitable, warm, with soft edges yet resistant, with boundaries yet suggesting a continuous safe expanse, a field that could be bundled, a bundle that could be unfurled, portable equipment, light, washable, long-lasting, colorful, versatile, functional and ornamental, private and universal, mine and thine.”
—Radka Donnell-Vogt, U.S. quiltmaker. As quoted in Lives and Works, by Lynn F. Miller and Sally S. Swenson (1981)
“I would have broke mine eye-strings, cracked them, but
To look upon him, till the diminution
Of space had pointed him sharp as my needle;
Nay, followed him till he had melted from
The smallness of a gnat to air, and then
Have turned mine eye and wept.”
—William Shakespeare (1564–1616)