Some results on the syzygies of finite sets and algebraic
... (0.2). Given a vector space V of dimension r + 1 over k, S = Sym(V) denotes the symmetric algebra on V, so that S is isomorphic to the polynomial ring k[xo, jcj. We denote by k the residue field S/(x0, ..., Xr) of S at the irrelevant maximal ideal. For a graded S-module T, we write T for its compone ...
... (0.2). Given a vector space V of dimension r + 1 over k, S = Sym(V) denotes the symmetric algebra on V, so that S is isomorphic to the polynomial ring k[xo, jcj. We denote by k the residue field S/(x0, ..., Xr) of S at the irrelevant maximal ideal. For a graded S-module T, we write T for its compone ...
ALGEBRAIC GEOMETRY - University of Chicago Math
... (b) Prove that C is smooth at every one of its points if and only if the equation for C is irreducible if and only if its equation can be put in the first form of (a). (c) Prove that C is singular at every one of its points if and only if the equation for C can be put in the third form of (a). (d) P ...
... (b) Prove that C is smooth at every one of its points if and only if the equation for C is irreducible if and only if its equation can be put in the first form of (a). (c) Prove that C is singular at every one of its points if and only if the equation for C can be put in the third form of (a). (d) P ...
9. Algebraic versus analytic geometry An analytic variety is defined
... where OUan (U ) is the ring of holomorphic functions on U . an Globally, we have a locally ringed space (X, OX ), where X is locally isomorphic to an analytic closed subset of some open subset U ⊂ Cn together with its sheaf of analytic functions. Theorem 9.1 (Chow’s Theorem). Let X ⊂ Pn be a closed ...
... where OUan (U ) is the ring of holomorphic functions on U . an Globally, we have a locally ringed space (X, OX ), where X is locally isomorphic to an analytic closed subset of some open subset U ⊂ Cn together with its sheaf of analytic functions. Theorem 9.1 (Chow’s Theorem). Let X ⊂ Pn be a closed ...
Regular differential forms
... 2 . It follows that there are no regular differential forms on P . Example Take the projective curve Y 2 Z = X 3 + XZ 2 . The projective plane P2 is covered by three affine pieces: A1 is the part with Z 6= 0 and coordinates (X, Y, 1), A2 is the part with Y 6= 0 and coordinates (U, 1, V ), A3 is the ...
... 2 . It follows that there are no regular differential forms on P . Example Take the projective curve Y 2 Z = X 3 + XZ 2 . The projective plane P2 is covered by three affine pieces: A1 is the part with Z 6= 0 and coordinates (X, Y, 1), A2 is the part with Y 6= 0 and coordinates (U, 1, V ), A3 is the ...
15. The functor of points and the Hilbert scheme Clearly a scheme
... The corresponding scheme is called the Hilbert scheme. For example, consider plane curves of degree d. The component of the Hilbert scheme is particularly nice in these examples, it is just represented by a projective space of dimension ...
... The corresponding scheme is called the Hilbert scheme. For example, consider plane curves of degree d. The component of the Hilbert scheme is particularly nice in these examples, it is just represented by a projective space of dimension ...
H10
... Remark. One can also show that every finitely generated flat module is projective, so for finitely generated A-modules flat and projective are equivalent. However, there are (non-finitely generated) flat modules which are not projective. For example, the ring Z[ 12 ] is a flat module over Z, but not ...
... Remark. One can also show that every finitely generated flat module is projective, so for finitely generated A-modules flat and projective are equivalent. However, there are (non-finitely generated) flat modules which are not projective. For example, the ring Z[ 12 ] is a flat module over Z, but not ...
Homework sheet 6
... 4. If X is a topological space, {Ui }i∈I is an open cover of X, and Z is a subset of X, prove that Z is closed if and only if Z ∩ Ui is closed in Ui (when Ui is given the induced topology) for all i. [This verifies a claim made in class.] 5. Let C be a smooth projective plane curve. Let ` be a fixed ...
... 4. If X is a topological space, {Ui }i∈I is an open cover of X, and Z is a subset of X, prove that Z is closed if and only if Z ∩ Ui is closed in Ui (when Ui is given the induced topology) for all i. [This verifies a claim made in class.] 5. Let C be a smooth projective plane curve. Let ` be a fixed ...
Natural Homogeneous Coordinates
... And Cartesian Coordinate Pairs • Let z = 1. – Then a projective coordinate line given by ...
... And Cartesian Coordinate Pairs • Let z = 1. – Then a projective coordinate line given by ...
索书号：O187 /C877 (2) (MIT) Ideals, Varieties, and Algorithms C
... Algebraic Geometry is the study of systems of polynomial equations in one or more variables, asking such questions as: Does the system have finitely many solutions, and if so how can one find them? And if there are infinitely many solutions, how can they be described and manipulated? The solution of ...
... Algebraic Geometry is the study of systems of polynomial equations in one or more variables, asking such questions as: Does the system have finitely many solutions, and if so how can one find them? And if there are infinitely many solutions, how can they be described and manipulated? The solution of ...
OX(D) (or O(D)) for a Cartier divisor D on a scheme X (1) on
... Recall that an eective Cartier divisor D is de ned, locally on some open a ne U = Spec(A), by a nonzerodivisor equation f ∈ A up to units in A (cf Hartshorne p. 141 and 145). To it, is associated the invertible sheaf OX (D) de ned on U as the module f1 A. Remark 1 : in this general de nition of OX ...
... Recall that an eective Cartier divisor D is de ned, locally on some open a ne U = Spec(A), by a nonzerodivisor equation f ∈ A up to units in A (cf Hartshorne p. 141 and 145). To it, is associated the invertible sheaf OX (D) de ned on U as the module f1 A. Remark 1 : in this general de nition of OX ...