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Pol Pol Pol...
and the symbolic expression of a function as an expression in . Example 1.3. Let . In this case Pol consists of quadratic forms in two variables . The discriminant is an obvious invariant of SL . We have...
It follows from the definition that the symbolic expression of any invariant polynomial from Pol Pol is multilinear. Let us show that it is also multiisobaric: Proposition 1.1.
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(i) Show that the function is multihomogeneous of multi-degree . (ii) Show that pol . 1.9 Find the symbolic expression for the polynomial on . Show that it is an invariant for the group the space of binary quartics Pol SL ....
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Thus, letting to get run through the set
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where denotes a column vector with coordinates equal to . Let length of this vector. It is clear that...
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Exercises
2.2 Let be the omega-operator in the polynomial ring . Prove that (i) for negative integers , (ii) , is a solution of the differential equation (iii) the function in the ring of formal power series .
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There are also some simple groups of exceptional type of types . Every simple algebraic group is isogeneous to one of these groups (i.e., there exists a surjective homomorphism from one to another with a finite kernel). We shall start the proof of Nagata’s Theorem with the following....
Proof. (i) (ii) Let and let Specm . is an irreducible algebraic variety on which acts (via the action of on ). Consider the canonical morphism such that is is isomorphic to an open subset of an affine variety , the identity. Since the restriction map defines a morphism of affine
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Proof. To simplify the proof let us assume that . We shall also identify with its image in GL ; which is isomorphic to . This can be done since
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In particular, we obtain that is an -torsion element if and only if is the divisor of a rational function. Let us now take . Assume that there exists a polynomial as in the statement of the lemma. Let be the equation of the inflection tangent at the point . Then the restriction of the rational function on to the curve defines a rational function with . Thus is an -torsion element in the group law. Conversely, assume that the latter occurs. By the above there exists a rational function with . By changing the projective coordinates if necessary, we may assume that the equation of is and that none of the points is the point with projective coordinates . Then the rational function is regular on the affine curve . Hence it can be represented by a polynomial with nonzero constant term. Homogenizing this polynomial, we obtain a homogeneous polynomial which is not divisible by such that the curve cuts out the divisor . By Bezout’s Theorem, ´ the degree of is equal to . Note that is not defined uniquely since we can always add to it a polynomial of the form , where is a homogeneous polynomial of degree . The rational function cuts out the same divisor on . Now we have to show that can be chosen in such a way that has multiplicity at each point . Let be the local ring of at the point and let be its maximal ideal. Since was assumed to be nonsingular, one can find a system of generators of such that is a local equation of at . We shall identify the formal completion of with the ring of formal power series in such a way that under the inclusion the image...
Proof. Let be the space of homogeneous polynomials of degree in . As we explained in the proof of Lemma 4.5, the dimension of this space is greater than or equal to . Thus we see that . In view of our assumption we must have . Since again by assumption we see that for sufficiently large ,...
given by homogeneous polynomials of degree . This map is SL -equivariant with respect to the natural actions of SL on the domain and the target space....
Proof. Without loss of generality, we may assume that and is an irredoes not contain any nontrivial proper submodules. ducible submodule, i.e., Consider the natural map of -modules Hom Hom . By...
Let be a linearly reductive connected affine algebraic group and let GL be its rational linear representation. Let be a maximal unipotent subgroup of a connected linearly reductive group . The reader unfamiliar with the...
Using the previous remark this has a simple geometric interpretation. In this case the Veronese variety is a conic, and the kernel of is one-dimensional. It is spanned by a quadratic polynomial vanishing on the conic.
r Y(...
Bibliographical notes
The notion of a covariant of a quantic (i.e., a homogeneous form) goes back to A. Cayley. It is discussed in all classical books in invariant theory. The fact that a covariant of a binary form corresponds to a semiinvariant was first discovered by M. Roberts in 1861 ([91]). It can already be found in Salmon’s book [97]. The result that the algebra of covariants of a binary form is finitely generated was first proved by P. Gordan [38] (see also classical proofs in [28], [39]). A modern proof SL on the can be found in [113]. Theorem 5.4 applied to the action of algebra Pol Pol is a generalization of Gordan’s Theorem. The first proof of this theorem was given by M. Khadzhiev [61]. Our exposition of the modern theory of covariants follows [89]. The algebra of covariants of binary forms of degree was computed by P. Gordan for ([38]) and by F. von Gall for degree ([36], [35]) (the proof of completeness of the generating set for may not be correct). For ternary forms the computations are known only for forms of degree 3 ([37], [42]) and incomplete for degree 4 ([98], [19]) (a thesis of Emmy Noether was devoted to such computations). Combinants of two binary forms of degrees are known in the cases ([96]; see a modern in [81]). Also known are combinants of two account of the case ternary forms of degrees ([28])....
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