I am looking for a reference on complex algebraic projective geometry. Specifically, I would like to become more acquainted with notions like the dimension and the degree of a projective algebraic set, intersection theory (Bézout's theorem), etc. For example:

Let $V,W$ be algebraic hypersurfaces of a complex projective space, and assume that $V \subset W$. Assume moreover that $V=\{P=0\}$ and $W=\{Q=0\}$, where $P,Q$ are homogenous polynomials, and $\mathrm{deg} P \geq \mathrm{deg} Q$.

I can see that this should imply $V=W$. However, I don't want to reinvent the wheel, and I would like to see this kind of things treated systematically, if possible without using too much machinery (like schemes).

  • $\begingroup$ Maybe Mumford's Algebraic Geometry I: Complex Projective Varieties would do? $\endgroup$ – Hoot Jan 14 '16 at 23:18

There's a bunch of references. Let me cite a few from which I've learnt the basics

Griffihs & Harris - Principles of Algebraic Geometry

Huybrechts - Complex Geometry, An introduction

Shafarevich - Basic Algebraic Geometry 1, 2

Fulton - Algebraic Curves

Hartshorne - Algebraic Geometry

Demailly - Complex Analytic and Differential Geometry

Some of these are far from discuss only the basics or only the complex case but they cover all that I think you'll need for a while.

  • $\begingroup$ thanks ! any one that you would recommend specifically? $\endgroup$ – hal Jan 14 '16 at 17:10
  • $\begingroup$ Fulton for the first steps. Then Shafarevich and Huybrechts (if you want a differential geometry approach). $\endgroup$ – Alan Muniz Jan 14 '16 at 17:13
  • $\begingroup$ great, thanks ! $\endgroup$ – hal Jan 14 '16 at 17:14
  • $\begingroup$ It would help if you study something about Several Complex Variables and Riemann Surfaces. $\endgroup$ – Alan Muniz Jan 14 '16 at 17:15
  • $\begingroup$ I do have a background in riemann surfaces and teichmüller theory, and a little bit of SVC. my main problem is that I am very ignorant in algebra $\endgroup$ – hal Jan 14 '16 at 17:16

Intersect with a copy of an affine space, so now $P, Q$ are inhomogeneous polynomials $p, q$, but choose the copy of affine space so that $p, q$ continue to have the same degrees as $P, Q$. If $V \subseteq W$, then the polynomial $q$ vanishes on $V$, so by the Nullstellensatz some power of $q$ lies in the ideal $(p)$ generated by $p$.

At this point you need some additional hypothesis on $p$ to rule out annoying possibilities like $p = x^2, q = xy$. Let me assume that $p$ is irreducible. Then $p$ divides $q$, so $q$ is a scalar multiple of $q$ and $V = W$ as desired.

  • $\begingroup$ thanks, but I included this specific question as an example of the kind of questions I would like to be more familiar with. do you have any book to suggest to that effect? $\endgroup$ – hal Jan 14 '16 at 16:50

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