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Name of book, while this is on the front page
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LSpice
LSpice
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I can also recommend Dolgachev's book Dolgachev's bookLectures on invariant theory. It used be freely available on his home page, but now it is published. It treats both some of the classical invariant theory and GIT.

The difference between the two is the following. In classical IT you are interested in finding the invariants of a ring under the action of a group. The prototypical example is the description of the algebra of symmetric polynomials as the polynomial algebra on elementary symmetric polynomials.

In GIT you do the following. Let $A$ be a ring, maybe the function ring of some affine algebraic variety, with an action of $G$. Then $A^G$ should be the ring of functions on the quotient of your variety by $G$. So invariant theory is viewed as a tool to describe the function rings of quotient varieties. The problem is that not all varieties are affine, and so GIT goes on to study what does it mean to take the quotient by the action of a group of a more general variety (or scheme), tipicallytypically in the projective case. One of the main difference is that it turns out that there are bad points that you have to discard altogether before taking a quotient.

I can also recommend Dolgachev's book. It used be freely available on his home page, but now it is published. It treats both some of the classical invariant theory and GIT.

The difference between the two is the following. In classical IT you are interested in finding the invariants of a ring under the action of a group. The prototypical example is the description of the algebra of symmetric polynomials as the polynomial algebra on elementary symmetric polynomials.

In GIT you do the following. Let $A$ be a ring, maybe the function ring of some affine algebraic variety, with an action of $G$. Then $A^G$ should be the ring of functions on the quotient of your variety by $G$. So invariant theory is viewed as a tool to describe the function rings of quotient varieties. The problem is that not all varieties are affine, and so GIT goes on to study what does it mean to take the quotient by the action of a group of a more general variety (or scheme), tipically in the projective case. One of the main difference is that it turns out that there are bad points that you have to discard altogether before taking a quotient.

I can also recommend Dolgachev's book Lectures on invariant theory. It used be freely available on his home page, but now it is published. It treats both some of the classical invariant theory and GIT.

The difference between the two is the following. In classical IT you are interested in finding the invariants of a ring under the action of a group. The prototypical example is the description of the algebra of symmetric polynomials as the polynomial algebra on elementary symmetric polynomials.

In GIT you do the following. Let $A$ be a ring, maybe the function ring of some affine algebraic variety, with an action of $G$. Then $A^G$ should be the ring of functions on the quotient of your variety by $G$. So invariant theory is viewed as a tool to describe the function rings of quotient varieties. The problem is that not all varieties are affine, and so GIT goes on to study what does it mean to take the quotient by the action of a group of a more general variety (or scheme), typically in the projective case. One of the main difference is that it turns out that there are bad points that you have to discard altogether before taking a quotient.

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Andrea Ferretti
Andrea Ferretti
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I can also recommend Dolgachev's book. It used be freely available on his home page, but now it is published. It treats both some of the classical invariant theory and GIT.

The difference between the two is the following. In classical IT you are interested in finding the invariants of a ring under the action of a group. The prototypical example is the description of the algebra of symmetric polynomials as the polynomial algebra on elementary symmetric polynomials.

In GIT you do the following. Let $A$ be a ring, maybe the function ring of some affine algebraic variety, with an action of $G$. Then $A^G$ should be the ring of functions on the quotient of your variety by $G$. So invariant theory is viewed as a tool to describe the function rings of quotient varieties. The problem is that not all varieties are affine, and so GIT goes on to study what does it mean to take the quotient by the action of a group of a more general variety (or scheme), tipically in the projective case. One of the main difference is that it turns out that there are bad points that you have to discard altogether before taking a quotient.