appendix [ag-000B]
✍️source

For draft notes, see Drafts for Notes on Algebraic Geometry.

Context

Notes on Algebraic Geometry [ag-0001]

August 13, 2024
Utensil Song

What is Algebraic Geometry? [ag-0005]

Algebraic geometry deals with solution sets of systems of polynomial equations [borisov2024adventures, sec. 1] from an geometric view, with many algebraic concepts imitating the notions of analysis and topology [kriz2021introduction, p. vii], but has the advantage of being able to deal with singularities [mehrle2017algebraic, lect. 1, sec. 1] and other pathological objects.

Algebraic geometry has developed in waves, each with its own language and point of view, see [hartshorne1977graduate, p. xiv] for some discussion. The older language is closer to the geometric intuition, while the newer language developes powerful techniques to solve problems in great generality, but the study of the latter is considered tedious, or even depressing, by many, when not accompanied with tangible applications [grothendieck1964elements, p. 12].

Remark. the grand plan [ag-0002]

We will use [kriz2021introduction] as a holistic guide to the organization of materials, which has done a great job of having the minimal prerequisites, being self-contained, and covering most of the topics that concern us, including the geometric motivation.

For a similar purpose, we use the formalization papers [bordg2022simple] and [buzzard2022schemes] to guide the path, at least the part towards schemes, and their counterparts in the Mathlib of Lean 4.

For prerequisites in basic algebra and commutative algebra, we will use [knapp2006basic] and [knapp2007advanced], and notes by Andreas Gathmann [gathmann2023plane][gathmann2013commutative] and David Mehrle [mehrle2015commutative]. For the geometric motivation and intuition, we will use [cox1997ideals] and [borisov2024adventures].

For upstream treatment of algebraic geometry, we will use [grothendieck1964elements] (particularly the English translation available at ryankeleti/ega) and [fantechi2006fundamental]. For modern notes, we will use [vakil2024rising], [gathmann2022algebraic] and [mehrle2017algebraic], with an eye on the classic textbook [hartshorne1977graduate].

We also need to tap into the language of Stacks in a modern setting, as treated in [khan2023lectures], with preliminaries on $\infty $-categories and derived categories.

See the plan for notes on algebraic geometry for an early discussion of the plan.

Convention. rings and fields [gathmann2013commutative, 0.1] [ag-0008]

A ring, usually denoted $R$, is always assumed to be a commutative ring (i.e. $a+b=b+a$ and $a b=b a$ for all $ a, b \in R$) with 1 (i.e. the multiplicative identity element, or called multiplicative unit).

$1 \neq 0$ is not required, where $0$ is the additive neutral element. If $1=0$, then $R$ must be the zero ring (or called the trivial ring), which consisting of one element, and is denoted $\{0\}$.

Subrings must have the same unit, and ring homomorphisms are always required to map $1$ to $1$.

A field, usually denoted $K$, is a commutative ring with $1$, where every nonzero element has a multiplicative inverse (thus division can be defined).

Definition. formal variable, formal expression [ag-0009]

A formal variable is an arbitrary symbol that is used to represent some mathematical object, and assumes nothing about the value or nature of the object.

A formal expression is mathematical expression with formal variables, assuming nothing but the formal variables participate in the expression have operations that are used in the expression.

A formal expression can be evaluated by replacing the formal variables with actual mathematical objects that have the operations defined in the expression.

When there is no ambiguity, we may omit the word "formal" and simply say variables or an expression.

Definition. monomial [cox1997ideals, 1.1.1] [ag-0003]

A monomial in $n$ formal variables $x_1, \ldots , x_n$, denoted $x^\alpha $, is a formal expression of the form \[ x_1^{\alpha _1} x_2^{\alpha _2} \cdots x_n^{\alpha _n} \] where $n \in \mathbb N$, $\alpha =\left (\alpha _1, \ldots , \alpha _n\right )$ is an $n$-tuple of nonnegative integers.

The total degree of the monomial is denoted $|\alpha |=\alpha _1+\cdots +\alpha _n$.

Definition. polynomial [cox1997ideals, 1.1.2, 1.1.3] [ag-0004]

A polynomial $f$ over a ring $R$ in $n$ variables is a finite linear combination (with coefficients $a_\alpha $ in $R$ ) of monomials, i.e. the formal expression of the form \[ f=\sum _\alpha a_\alpha x^\alpha , \quad a_\alpha \in R, \]

The set of all polynomials in $x_1, \ldots , x_n$ with coefficients in $R$ is denoted $R\left [x_1, \ldots , x_n\right ]$.

$a_\alpha x^\alpha $ is called a term of $f$ if $a_\alpha \neq 0$.

The total degree of $f \neq 0$, denoted $\operatorname {deg}(f)$, is the maximum $|\alpha |$ such that the coefficient $a_\alpha $ is nonzero. The total degree of the zero polynomial is undefined.

Remark. polynomial ring [cox1997ideals, 1.1.3] [ag-0007]

Under addition and multiplication, $R\left [x_1, \ldots , x_n\right ]$ satisfies all axioms of a commutative ring, and for this reason we will refer to $R\left [x_1, \ldots , x_n\right ]$ as a polynomial ring.

Definition. affine space [gathmann2013commutative, 0.3] [ag-000A]

The ($n$-dimensional) affine space over a field $K$, denoted $\mathbb {A}_K^n$, is \[ \left \{\left (c_1, \ldots , c_n\right ): c_i \in K \text { for } i=1, \ldots , n\right \} \] which is just $K^n$ as a set, without the its additional structures as a $K$-vector space and a ring.

We'll often use the term affine $n$-space to indicate the dimension. Particularly, an affine 1-space is called an affine line, an affine 2-space is an affine plane.

Definition. affine variety [gathmann2013commutative, 0.3] [ag-000D]

Let $S \subset K\left [x_1, \ldots , x_n\right ]$ be a set of polynomials. The zero locus (or zero set) of $S$ is \[ V(S):=\left \{x \in \mathbb {A}_K^n: f(x)=0 \text { for all } f \in S\right \} \subset \mathbb {A}_K^n \]

An affine algebraic variety over $K$ is a subset of $\mathbb {A}_K^n$ of this form. It's usually simply called an affine variety over $K$, or an affine $K$-variety.

If $S=\left (f_1, \ldots , f_k\right )$ is a finite set, $V(S)$ can be written as $V\left (f_1, \ldots , f_k\right )$.

Obviously, it is the set of all solutions of the system of polynomial equations $f_1\left (x_1, \ldots , x_n\right )=\cdots =f_s\left (x_1, \ldots , x_n\right )=0$ [cox1997ideals, 1.2.1], denoted $\operatorname {Sol}(S;K)$ [dolgachev2013introduction, p. 1].

Remark. affine v.s. projective [michalek2021invitation, ch. 2] [ag-000E]

The prefix "affine" of affine variety is used to distinguish it from a projective variety. Affine varieties arise from arbitrary polynomials, while projective varieties arise from systems of homogeneous polynomials, i.e. linear combinations of monomials of fixed degree.

Since affine varieties are the general case, they are sometimes simply called varieties.

Figure out its relation to affine space in Geometry, which preserves parallelism and ratio of lengths for parallel line segments, but not distances and measures of angles.

Example. varieties [ag-000F]

$\mathbf {V}\left (10 x^2-x^3-y^2\right )$ from [cox1997ideals, p. 24]:

$\text{[math]}$

$\mathbf {V}\left (x^2-y^2 z^2+z^3\right )$ from [cox1997ideals, p. 7, p. 16]:

$\text{[math]}$

Figure. Implicit surfaces [uts-000J]

#define AS_LIB 1

int get_shape() {
    return int(iTime) % 52;
}

#include "/forest/shader/implicit.glsl"

void mainImage( out vec4 fragColor, in vec2 fragCoord )
{
    vec2 uv = 2.*(fragCoord-iResolution.xy/2.)/iResolution.y; // contains [-1,1]^2
    vec3 col = vec3(0.);
    
    // Camera rays
    vec3 camPos = vec3(4.,0.,0.);
    vec3 camDir = - normalize(camPos);
    vec3 rayPos, rayDir;
    float zoom = 1.3; // 1.8*cos(iTime);
    // if (checkKey(KEY_E)) zoom = 0.5;
    float fov = 0.4*zoom;
    float fov_ortho = 1.5*zoom;
    #if perspective
        // perspective cam
        rayPos = camPos;
        rayDir = normalize(camDir + fov*vec3(0., uv.x, uv.y));
    #else
        // orthographic cam
        rayPos = camPos + fov_ortho*vec3(0., uv.x, uv.y);
        rayDir = camDir;
    #endif
    // for perspective background in orthographic mode
    vec3 cubemapDir = normalize(camDir + fov*vec3(0., uv.x, uv.y));
    
    // Mouse-controlled rotation
    vec2 mouse = initMouse + vec2(0.015625*sin(iTime*PI), 0.0); // initMouse; // iMouse.xy == vec2(0.,0.) ? initMouse : (iMouse.xy/iResolution.xy - 0.5);
    float yaw = clamp(- mouse.x * 2.*PI * 1., -PI,PI);
    float pitch = clamp( mouse.y * PI * 1.2, -PI*0.5, PI*0.5);
    // pitch and yaw rotations (column-wise matrices)
    mat3 rot = mat3(cos(yaw), sin(yaw), 0., -sin(yaw), cos(yaw), 0., 0., 0., 1.);
    rot = rot * mat3(cos(pitch), 0., -sin(pitch), 0., 1., 0., sin(pitch), 0., cos(pitch));
    // apply
    camPos = rot*camPos;
    camDir = rot*camDir;
    rayPos = rot*rayPos;
    rayDir = rot*rayDir;
    cubemapDir = rot*cubemapDir;
    
    //cubemapDir = vec3(cubemapDir.x, cubemapDir.z, cubemapDir.y);
    
    vec3 hitPoint = raycast(rayPos, rayDir);
    
    if (hitPoint == BINGO) { fragColor = vec4(BINGO,1.0); return; }
    //if (hitPoint == NOHIT) { fragColor = vec4(NOHIT,1.0); return; }
    //if (hitPoint == NOBOUNDHIT) { fragColor = vec4(NOBOUNDHIT,1.0); return; }
    //if (hitPoint == ESCAPEDBOUNDS) { fragColor = vec4(ESCAPEDBOUNDS,1.0); return; }
    //if (hitPoint == MAXDISTREACHED) { fragColor = vec4(MAXDISTREACHED,1.0); return; }
    //if (hitPoint == MAXITERREACHED) { fragColor = vec4(MAXITERREACHED,1.0); return; }
    
    if (hitPoint == NOBOUNDHIT  || hitPoint == NOHIT || hitPoint == ESCAPEDBOUNDS || hitPoint == MAXITERREACHED) {
        //fragColor = vec4(vec3(0.2),1.0); return;

        // make background transparent
        fragColor = vec4(0.0,0.0,0.0,0.0); return;
        
        col = with_background(cubemapDir);
        
        #if showBoundingCube
        // darken bounding cube
        if (hitPoint != NOBOUNDHIT) { col *= vec3(0.7); }
        #endif

        fragColor = vec4(col,1.0); return;
    }

    vec3 grad = gradf(hitPoint+1.1*EPS*(-rayDir));
    float s = -sign(dot(grad,rayDir));
    
    col = with_color_mode(grad, s, hitPoint, camPos);

    col = clamp(col, 0., 1.);

    col = with_surface_pattern(col, hitPoint);

    col = with_shading(col, grad, s, rayDir);
    
    col = clamp(col, 0., 1.);

    fragColor = vec4(col,1.0);
}

adapted from Rogolop's Implicit surfaces

appendix [ag-000B]
✍️source

For draft notes, see Drafts for Notes on Algebraic Geometry.

Drafts for Notes on Algebraic Geometry [ag-000C]

Notes on Algebraic Geometry [ag-0001]

August 13, 2024
Utensil Song

What is Algebraic Geometry? [ag-0005]

Algebraic geometry deals with solution sets of systems of polynomial equations [borisov2024adventures, sec. 1] from an geometric view, with many algebraic concepts imitating the notions of analysis and topology [kriz2021introduction, p. vii], but has the advantage of being able to deal with singularities [mehrle2017algebraic, lect. 1, sec. 1] and other pathological objects.

Algebraic geometry has developed in waves, each with its own language and point of view, see [hartshorne1977graduate, p. xiv] for some discussion. The older language is closer to the geometric intuition, while the newer language developes powerful techniques to solve problems in great generality, but the study of the latter is considered tedious, or even depressing, by many, when not accompanied with tangible applications [grothendieck1964elements, p. 12].

Remark. the grand plan [ag-0002]

We will use [kriz2021introduction] as a holistic guide to the organization of materials, which has done a great job of having the minimal prerequisites, being self-contained, and covering most of the topics that concern us, including the geometric motivation.

For a similar purpose, we use the formalization papers [bordg2022simple] and [buzzard2022schemes] to guide the path, at least the part towards schemes, and their counterparts in the Mathlib of Lean 4.

For prerequisites in basic algebra and commutative algebra, we will use [knapp2006basic] and [knapp2007advanced], and notes by Andreas Gathmann [gathmann2023plane][gathmann2013commutative] and David Mehrle [mehrle2015commutative]. For the geometric motivation and intuition, we will use [cox1997ideals] and [borisov2024adventures].

For upstream treatment of algebraic geometry, we will use [grothendieck1964elements] (particularly the English translation available at ryankeleti/ega) and [fantechi2006fundamental]. For modern notes, we will use [vakil2024rising], [gathmann2022algebraic] and [mehrle2017algebraic], with an eye on the classic textbook [hartshorne1977graduate].

We also need to tap into the language of Stacks in a modern setting, as treated in [khan2023lectures], with preliminaries on $\infty $-categories and derived categories.

See the plan for notes on algebraic geometry for an early discussion of the plan.

Convention. rings and fields [gathmann2013commutative, 0.1] [ag-0008]

A ring, usually denoted $R$, is always assumed to be a commutative ring (i.e. $a+b=b+a$ and $a b=b a$ for all $ a, b \in R$) with 1 (i.e. the multiplicative identity element, or called multiplicative unit).

$1 \neq 0$ is not required, where $0$ is the additive neutral element. If $1=0$, then $R$ must be the zero ring (or called the trivial ring), which consisting of one element, and is denoted $\{0\}$.

Subrings must have the same unit, and ring homomorphisms are always required to map $1$ to $1$.

A field, usually denoted $K$, is a commutative ring with $1$, where every nonzero element has a multiplicative inverse (thus division can be defined).

Definition. formal variable, formal expression [ag-0009]

A formal variable is an arbitrary symbol that is used to represent some mathematical object, and assumes nothing about the value or nature of the object.

A formal expression is mathematical expression with formal variables, assuming nothing but the formal variables participate in the expression have operations that are used in the expression.

A formal expression can be evaluated by replacing the formal variables with actual mathematical objects that have the operations defined in the expression.

When there is no ambiguity, we may omit the word "formal" and simply say variables or an expression.

Definition. monomial [cox1997ideals, 1.1.1] [ag-0003]

A monomial in $n$ formal variables $x_1, \ldots , x_n$, denoted $x^\alpha $, is a formal expression of the form \[ x_1^{\alpha _1} x_2^{\alpha _2} \cdots x_n^{\alpha _n} \] where $n \in \mathbb N$, $\alpha =\left (\alpha _1, \ldots , \alpha _n\right )$ is an $n$-tuple of nonnegative integers.

The total degree of the monomial is denoted $|\alpha |=\alpha _1+\cdots +\alpha _n$.

Definition. polynomial [cox1997ideals, 1.1.2, 1.1.3] [ag-0004]

A polynomial $f$ over a ring $R$ in $n$ variables is a finite linear combination (with coefficients $a_\alpha $ in $R$ ) of monomials, i.e. the formal expression of the form \[ f=\sum _\alpha a_\alpha x^\alpha , \quad a_\alpha \in R, \]

The set of all polynomials in $x_1, \ldots , x_n$ with coefficients in $R$ is denoted $R\left [x_1, \ldots , x_n\right ]$.

$a_\alpha x^\alpha $ is called a term of $f$ if $a_\alpha \neq 0$.

The total degree of $f \neq 0$, denoted $\operatorname {deg}(f)$, is the maximum $|\alpha |$ such that the coefficient $a_\alpha $ is nonzero. The total degree of the zero polynomial is undefined.

Remark. polynomial ring [cox1997ideals, 1.1.3] [ag-0007]

Under addition and multiplication, $R\left [x_1, \ldots , x_n\right ]$ satisfies all axioms of a commutative ring, and for this reason we will refer to $R\left [x_1, \ldots , x_n\right ]$ as a polynomial ring.

Definition. affine space [gathmann2013commutative, 0.3] [ag-000A]

The ($n$-dimensional) affine space over a field $K$, denoted $\mathbb {A}_K^n$, is \[ \left \{\left (c_1, \ldots , c_n\right ): c_i \in K \text { for } i=1, \ldots , n\right \} \] which is just $K^n$ as a set, without the its additional structures as a $K$-vector space and a ring.

We'll often use the term affine $n$-space to indicate the dimension. Particularly, an affine 1-space is called an affine line, an affine 2-space is an affine plane.

Definition. affine variety [gathmann2013commutative, 0.3] [ag-000D]

Let $S \subset K\left [x_1, \ldots , x_n\right ]$ be a set of polynomials. The zero locus (or zero set) of $S$ is \[ V(S):=\left \{x \in \mathbb {A}_K^n: f(x)=0 \text { for all } f \in S\right \} \subset \mathbb {A}_K^n \]

An affine algebraic variety over $K$ is a subset of $\mathbb {A}_K^n$ of this form. It's usually simply called an affine variety over $K$, or an affine $K$-variety.

If $S=\left (f_1, \ldots , f_k\right )$ is a finite set, $V(S)$ can be written as $V\left (f_1, \ldots , f_k\right )$.

Obviously, it is the set of all solutions of the system of polynomial equations $f_1\left (x_1, \ldots , x_n\right )=\cdots =f_s\left (x_1, \ldots , x_n\right )=0$ [cox1997ideals, 1.2.1], denoted $\operatorname {Sol}(S;K)$ [dolgachev2013introduction, p. 1].

Remark. affine v.s. projective [michalek2021invitation, ch. 2] [ag-000E]

The prefix "affine" of affine variety is used to distinguish it from a projective variety. Affine varieties arise from arbitrary polynomials, while projective varieties arise from systems of homogeneous polynomials, i.e. linear combinations of monomials of fixed degree.

Since affine varieties are the general case, they are sometimes simply called varieties.

Figure out its relation to affine space in Geometry, which preserves parallelism and ratio of lengths for parallel line segments, but not distances and measures of angles.

Example. varieties [ag-000F]

$\mathbf {V}\left (10 x^2-x^3-y^2\right )$ from [cox1997ideals, p. 24]:

$\text{[math]}$

$\mathbf {V}\left (x^2-y^2 z^2+z^3\right )$ from [cox1997ideals, p. 7, p. 16]:

$\text{[math]}$

Figure. Implicit surfaces [uts-000J]

#define AS_LIB 1

int get_shape() {
    return int(iTime) % 52;
}

#include "/forest/shader/implicit.glsl"

void mainImage( out vec4 fragColor, in vec2 fragCoord )
{
    vec2 uv = 2.*(fragCoord-iResolution.xy/2.)/iResolution.y; // contains [-1,1]^2
    vec3 col = vec3(0.);
    
    // Camera rays
    vec3 camPos = vec3(4.,0.,0.);
    vec3 camDir = - normalize(camPos);
    vec3 rayPos, rayDir;
    float zoom = 1.3; // 1.8*cos(iTime);
    // if (checkKey(KEY_E)) zoom = 0.5;
    float fov = 0.4*zoom;
    float fov_ortho = 1.5*zoom;
    #if perspective
        // perspective cam
        rayPos = camPos;
        rayDir = normalize(camDir + fov*vec3(0., uv.x, uv.y));
    #else
        // orthographic cam
        rayPos = camPos + fov_ortho*vec3(0., uv.x, uv.y);
        rayDir = camDir;
    #endif
    // for perspective background in orthographic mode
    vec3 cubemapDir = normalize(camDir + fov*vec3(0., uv.x, uv.y));
    
    // Mouse-controlled rotation
    vec2 mouse = initMouse + vec2(0.015625*sin(iTime*PI), 0.0); // initMouse; // iMouse.xy == vec2(0.,0.) ? initMouse : (iMouse.xy/iResolution.xy - 0.5);
    float yaw = clamp(- mouse.x * 2.*PI * 1., -PI,PI);
    float pitch = clamp( mouse.y * PI * 1.2, -PI*0.5, PI*0.5);
    // pitch and yaw rotations (column-wise matrices)
    mat3 rot = mat3(cos(yaw), sin(yaw), 0., -sin(yaw), cos(yaw), 0., 0., 0., 1.);
    rot = rot * mat3(cos(pitch), 0., -sin(pitch), 0., 1., 0., sin(pitch), 0., cos(pitch));
    // apply
    camPos = rot*camPos;
    camDir = rot*camDir;
    rayPos = rot*rayPos;
    rayDir = rot*rayDir;
    cubemapDir = rot*cubemapDir;
    
    //cubemapDir = vec3(cubemapDir.x, cubemapDir.z, cubemapDir.y);
    
    vec3 hitPoint = raycast(rayPos, rayDir);
    
    if (hitPoint == BINGO) { fragColor = vec4(BINGO,1.0); return; }
    //if (hitPoint == NOHIT) { fragColor = vec4(NOHIT,1.0); return; }
    //if (hitPoint == NOBOUNDHIT) { fragColor = vec4(NOBOUNDHIT,1.0); return; }
    //if (hitPoint == ESCAPEDBOUNDS) { fragColor = vec4(ESCAPEDBOUNDS,1.0); return; }
    //if (hitPoint == MAXDISTREACHED) { fragColor = vec4(MAXDISTREACHED,1.0); return; }
    //if (hitPoint == MAXITERREACHED) { fragColor = vec4(MAXITERREACHED,1.0); return; }
    
    if (hitPoint == NOBOUNDHIT  || hitPoint == NOHIT || hitPoint == ESCAPEDBOUNDS || hitPoint == MAXITERREACHED) {
        //fragColor = vec4(vec3(0.2),1.0); return;

        // make background transparent
        fragColor = vec4(0.0,0.0,0.0,0.0); return;
        
        col = with_background(cubemapDir);
        
        #if showBoundingCube
        // darken bounding cube
        if (hitPoint != NOBOUNDHIT) { col *= vec3(0.7); }
        #endif

        fragColor = vec4(col,1.0); return;
    }

    vec3 grad = gradf(hitPoint+1.1*EPS*(-rayDir));
    float s = -sign(dot(grad,rayDir));
    
    col = with_color_mode(grad, s, hitPoint, camPos);

    col = clamp(col, 0., 1.);

    col = with_surface_pattern(col, hitPoint);

    col = with_shading(col, grad, s, rayDir);
    
    col = clamp(col, 0., 1.);

    fragColor = vec4(col,1.0);
}

adapted from Rogolop's Implicit surfaces

appendix [ag-000B]
✍️source

For draft notes, see Drafts for Notes on Algebraic Geometry.

Definition. affine variety [cox1997ideals, 1.2.1] [ag-0006]

Let $f_1, \ldots , f_s$ be polynomials in $K\left [x_1, \ldots , x_n\right ]$. The affine variety defined by $f_1, \ldots , f_s$ is \[ \mathbf {V}\left (f_1, \ldots , f_s\right )=\left \{\left (a_1, \ldots , a_n\right ) \in \mathbb {A}_K^n \mid f_i\left (a_1, \ldots , a_n\right )=0 \text { for all } 1 \leq i \leq s\right \} \]

Obviously, it is the set of all solutions of the system of polynomial equations $f_1\left (x_1, \ldots , x_n\right )=\cdots =f_s\left (x_1, \ldots , x_n\right )=0$.

Definition. affine variety [gathmann2013commutative, 0.3] [ag-000D]

Let $S \subset K\left [x_1, \ldots , x_n\right ]$ be a set of polynomials. The zero locus (or zero set) of $S$ is \[ V(S):=\left \{x \in \mathbb {A}_K^n: f(x)=0 \text { for all } f \in S\right \} \subset \mathbb {A}_K^n \]

An affine algebraic variety over $K$ is a subset of $\mathbb {A}_K^n$ of this form. It's usually simply called an affine variety over $K$, or an affine $K$-variety.

If $S=\left (f_1, \ldots , f_k\right )$ is a finite set, $V(S)$ can be written as $V\left (f_1, \ldots , f_k\right )$.

Obviously, it is the set of all solutions of the system of polynomial equations $f_1\left (x_1, \ldots , x_n\right )=\cdots =f_s\left (x_1, \ldots , x_n\right )=0$ [cox1997ideals, 1.2.1], denoted $\operatorname {Sol}(S;K)$ [dolgachev2013introduction, p. 1].

the plan for notes on algebraic geometry [uts-000F]

The approach we would like to take is similar to the one we took in our Notes on Topos Theory and Type Theory: determine some goals, find some references that sketch out a path to those goals, and then write notes jumping between many references to fill in the gaps.

Goal 1: To be able to state "an affine scheme is a scheme", which means we need all the basics of prime ideals, Zariski topology, sheaves, spectrum of a ring, stalks, locally ringed spaces etc. as sketched out in [bordg2022simple] which has organized the minimum preliminaries in a formal way, but we might also need to refer to [buzzard2022schemes] and Mathlib to ensure that we can properly find the Lean counterparts.

Goal 2: To properly understand the geometric topic of algebraic geometry, we also need many basic correspondence in the Algebra-Geometry dictionary, which can be found in [cox1997ideals], including ideals, varieties, Gröbner Bases, elimination Theory, the relation between irreducible Varieties and prime Ideals, projective algebraic geometry etc. This book provides concrete intuitions and examples for working with the abstract concepts. We might also branch out from there to write notes about ray-tracing the implicit surfaces of algebraic varieties (see some links for rendering implicit surfaces for some links), and the use of Macaulay2 and Singular to do computations in AG.

Goal 3: A challenging goal is to be able to read EGA [grothendieck1964elements] and some materials surrounding it:

EGA: we will use the English translation available at ryankeleti/ega, with the French version (year 1960) available at Numdam, both use the word scheme to mean separated scheme, and prescheme to mean scheme. A Chinese translation is also available and has an index of terminology in Chinese, English, and French, which is useful. We would like to include terminology in all three languages for key concepts in our notes, showing only the English version by default.
FGA: we will use [fantechi2006fundamental] instead of the original FGA. This is an explained and extended version of FGA, that has the advantage of being in English and from a little more modern viewpoint.
SGA: Although topos is first introduced in SGA, but we are not ready to read SGA yet.
The rising sea: [vakil2024rising] is a modern introduction to algebraic geometry, used by many courses, e.g. math256 (which also has an English synopsis of EGA). Some courses, e.g. Algebraic Geometry Spring 2024 , also follows lecture notes by Andreas Gathmann [gathmann2022algebraic] (found on Math 8253: Algebraic Geometry).
FAC [serre1955faisceaux]: there is an English version Coherent Algebraic Sheaves translated by Piotr Achinger and Lukasz Krupa. This is no longer the modern treatment of AG, so it's only used as a reference.
GAGA [serre1956geometrie]: there is only the French version and the Chinese translation. The evaluation is the same as FAC.

We need some modern notes to avoid being lost in the old language, for that, besides The rising sea, we have chosen [borisov2024adventures] and [mehrle2017algebraic].

Goal 4: We also need to tap into the language of Stacks in a modern setting, as treated in [khan2023lectures], with preliminaries on $\infty $-categories and derived categories.