Right now I am reading David Mumford's book-Algebraic Geometry I Complex Projective Varieties, from the 3ed chapter, too much proofs lead me to lost in the book. I need more suitable for beginner so I can read David Mumford's book, that is with more examples and explanations. Or, some other books you think are suitable for me. I have some backgound on functional analysis, commutative algebra by A&M, Bertrametti et al. "Lectures on Curves, Surfaces and Projective Varieties".
Need some simpler books than Algebraic Geometry I Complex Projective Varieties by David Mumford
algebraic-geometrybook-recommendation
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The most appropriate answer will depend on why you are working through a book on Riemann surfaces and algebraic curves, but I will try to give some suggestions.
Since you mention Riemann surfaces, let's start with some analogy with smooth manifolds. The Whitney embedding theorem says that any smooth manifold can be embedded in $\mathbb{R}^N$ for $N$ sufficiently large, so we can always think of a smooth manifold as a submanifold of $\mathbb{R}^N$. This occasionally helps with intuition and visualization, and can simplify some constructions.
In the case of complex manifolds (e.g. Riemann surfaces), you might ask whether the same holds true holomorphically, i.e. whether any complex manifold can be holomorphically embedded in $\mathbb{C}^N$ for $N$ sufficiently large. It turns out that usually the answer is no. It is an easy consequence of the Liouville theorem that no compact complex manifold is a complex submanifold of $\mathbb{C}^N$. If you only care about compact complex manifolds, then $\mathbb{CP}^N$ turns out to be the best possible (see e.g. the Kodaira embedding theorem, which characterizes which compact complex manifolds are complex submanifolds of $\mathbb{CP}^N$).
If your motivation is the study of solutions to polynomial equations, then as mentioned in other answers and comments, projective spaces are the appropriate completions of affine space that allow as many solutions as possible, allowing various formulas (e.g. couting intersections) work without additional qualification.
About visualization: for curves in $\mathbb{CP}^2$, first take some affine chart $\mathbb{C}^2 \subset \mathbb{CP}^2$, and then look at the intersection with some "real slice" $\mathbb{R}^2 \subset \mathbb{C}^2$. For example if we look at the curve in $\mathbb{CP}^2$ given by the zero set of $x^2-yz$, by working on the affine chart $z\neq0$ this becomes $y = x^2$ on $\mathbb{C}^2$, and if we restrict to real $x,y$ we get a parabola.
Riemann surfaces is a very standard topics in math, then you can find a lot of books talking about Riemann surfaces under different point of views.
I can suggest you:
-Riemann Surfaces - S.Donaldson,
-Riemann Surfaces - Farkas and Kra,
-Algebraic curves and Riemann surfaces - R.Miranda
-Lectures on Riemann Surfaces - Otto Forster
Donaldson's book is more difficult with respect to others t, and he use a lot of basic algebraic geometry. I have read Forster's book and have been pretty impressed by it. Another excellent analytic monograph from this point of view is the Princeton lecture notes on Riemann surfaces by Robert Gunning, which is also a good place to learn sheaf theory. His main result is that all compact complex one manifolds occur as the Riemann surface of an algebraic curve. Miranda's book contains more study of the geometry of algebraic curves.
Riemann himself, as I recall, took an intermediate view, showing the equivalence of the categories of (irreducible) algebraic curves with that of (connected) compact complex manifolds equipped with a finite holomorphic map to P^1. Another extremely nice book, a little more advanced than Miranda, is the China notes on algebraic curves by Phillip Griffiths. Mumford's book Complex projective varieties I, also has a terrific chapter on curves from the complex analytic point of view.
After you learn the basics, the book of Arbarello, Cornalba, Griffiths, Harris, is just amazing. Of course Riemann's thesis and followup paper on theory of abelian functions is rather incredible as well.
Best Answer
An Invitation to Algebraic Geometry by Smith, K., Kahanpää, L., Kekäläinen, P., Traves
Is a great place to start, covers lots of material in an informal way, with lots of diagrams and intuitive explanations. It's perfect before starting a more formal study.
As mentioned in the comment by @jgon Milne has lecture notes on his site https://www.jmilne.org/math/ which provide an excellent introduction (as well as fantastic notes on a range of other subjects)