Index of terminology — EGA V

Alphabetized; first-occurrence reference. Synonym pairs are grouped under one entry with a cross-reference; Grothendieck marginal-note neologisms are tagged (Grothendieck note). Modern (post-EGA) alternative names — e.g. "scheme" vs. "prescheme", "Nagata ring" vs. "Japanese ring" — are listed separately when they appear in the source; otherwise the EGA-period term is the indexed form.

A

• absolute orthogonal group $O(n)$ — (V, 2.16.12)
• absolute projective group $PGL(r)$ — (V, 6.1.9)
• adjusted (corrected) discriminant of a quadratic form $\tilde{d}(Q)$ — (V, 2.16.6), (V, 2.16.7)
• adjusted discriminant polynomial (of the indeterminate quadratic form in $n$ variables) — (V, 2.16.5)
• augmentation of elementary quadratic type (case $n=2$ of elementary augmentation) — (V, 2.15.2)

B

• Bertini-Zariski theorem — (V, 5.3.1)
• birational morphism of $X$ onto a hypersurface — (V, 5.14.6)
• blow-up of $X$ with centre $T$ (linear-pencil interpretation) — (V, 5.10.2), (V, 5.10.3)
• Brauer-Severi schemes (twisted projective fibrations) — (V, 6) introduction; cf. (V, 6.4.3)

C

• canonical incidence divisor $H$ (on $P{\times }_S{P}^{\vee }$) — (V, 6.3.1); see also incidence prescheme
• classical / "old" language (vs. EGA terminology; en termes de papa) — (V, 1.7), (V, 5.8.12)
• codepth (coprof) — (V, 5.2.8), (V, 5.8.5)
• complete-intersection ring (a Noetherian local ring elementary of multiplicity $n$ is such) — (V, 2.15.6)
• conic projection of $X$ relative to $f$ with centre $C$ ($p_C^X$) — (V, 5.14)
• conic projection with centre $C$ ($p_C$) — (V, 5.14)
• conormal module ${\nu }_{X/P}^{\vee }$ — (V, 5.8.6)
• constructible property (of an algebraic prescheme over a field) — (V, 5.4)
• content of a polynomial (gcd of coefficients) — (V, 2.16.5)

D

• degree of an invertible module on a projective fibration (over a field) — (V, 6.1)
• degree of a family of divisors / linear system — (V, 6.3)
• degree of a morphism $g:P\to {\mathbb{P}}^1$ — (V, 6.1)
• degree of a relative divisor on a projective fibration — (V, 6.2.1)
• degree-$n$ morphism (or invertible module) on a projective fibration — (V, 6.1)
• determinant (Grothendieck's proposed renaming of the ordinary discriminant of a quadratic form) — (V, 2.16.11) (Grothendieck note)
• discriminant of a quadratic form $d(Q)$ — (V, 2.16.1); see also adjusted (corrected) discriminant
• double tangent (and dual curve) — (V, 5.8.10)

E

• elementary augmentation of multiplicity $n$ — (V, 2.15.2)
• elementary quadratic singularity (= ordinary singularity, classical sense) — (V, 2.15.2)
• elementary singular point of multiplicity $n$ — (V, 2.15.2)
• elementary singular zero of multiplicity $n$ — (V, 2.15.5)
• elementary singularity of multiplicity $n$ (local-ring sense) — (V, 2.15.2)
• étale local triviality (of a smooth quadratic form) — (V, 2.16.13)
• exceptional hyperplane (for a property $P$) — (V, 5.8)
• extended conic projection of $X$ relative to $f$ with centre $C$ — (V, 5.14)

F

• family of divisors over $X/S$ parametrized by $T$ — (V, 6.3.1)
• fixed component (of a family of divisors) — (V, 6.3)
• fixed point of a family of divisors — (V, 6.3.1)
• fixed point in the extended sense — (V, 6.3.1)
• $F$-regular section — (V, 5.5.2)

G

• generic hyperplane section — (V, 5.2), (V, 5.3)
• geometrically connected hyperplane section — (V, 5.6.1)
• geometrically elementary singularity of multiplicity $n$ — (V, 2.15.2)
• geometrically integral / geometrically irreducible (sense for hyperplane sections) — (V, 5.3.2), (V, 5.4.3)
• geometrically normal hyperplane section — (V, 5.4.3), (V, 5.8.14)
• geometrically singular (resp. supersingular) zero relative to $k$ — (V, 1.3)
• geometrically supersingular point (locally of finite type over a field) — (V, 2.15.1) (Grothendieck note)
• Grassmannian $Gras{s}_n(\mathcal{E})$ — (V, 5.11)

H

• hyperplane in $P$ defined by $\xi \in P^{\vee }(S)$ ($H_{\xi }$) — (V, 5.1)
• hyperplane section of $X$ (relative to a projective immersion and a hyperplane) — (V, 5.1)
• hyperplanes of $\mathbb{P}$ (codimension-one linear subvarieties) — (V, 5.11)

I

• incidence prescheme between $P$ and $P^{\vee }$ ($H$) — (V, 5.1); see also canonical incidence divisor
• incidence prescheme for linear sections ($H^{(m)}$) — (V, 5.12)
• invertible sheaf of degree $n$ — (V, 6.1)
• isomorphic linear systems of divisors — (V, 6.4.2)

L

• linear family of hyperplane sections passing through $x$ — (V, 5.9.3)
• linear morphism between projective fibrations — (V, 6.1)
• linear pencil of hyperplane sections of $X$ (Y_L) — (V, 5.10.1); see also pencils of hyperplane sections
• linear sections of $X$ over $\mathbb{P}$ by linear subvarieties of codimension $m$ ($Y_{\xi }^{(m)}$) — (V, 5.12)
• linear subvariety of $\mathbb{P}(\mathcal{E})$ — (V, 5.11)
• linear system of divisors on $X$ (set-of-divisors abuse of language) — (V, 6.4.5)
• linear system of divisors over $X/S$ parametrized by the projective fibration $Q=P^{\vee }$ — (V, 6.3)
• linear system without fixed components — (V, 6.3)
• linear system without fixed points — (V, 6.3)

M

• multidegree of an invertible module on a multiprojective fibration — (V, 6.2.6)
• multidegree of a relative divisor — (V, 6.2.7)
• multiprojective fibration $P={\prod }_i{P}_i$ — (V, 6.2.5)
• multisection of $X$ over $U$ — (V, 5.7.1)

N

• non-degenerate quadratic form — (V, 2.16.2)
• non-degenerate quadratic singularity — (V, 2.15.1) (Grothendieck note)
• non-singular zero — (V, 1.2)
• $n$-form on $E^{\vee }$ — (V, 2.15.1)

O

• ordinary quadratic singularity (classical) — (V, 2.15.2)
• ordinary quadratic zero / ordinary singular zero (smooth-form variant) — (V, 2.15.1) (Grothendieck note)
• ordinary singular zero (classical, §V.1) — (V, 1.1)
• orthogonal-group fibration (the principal homogeneous fibration $Isom(E_0,E)\to \mathcal{Q}(E)\ast$) — (V, 2.16.12)
• orthogonal group scheme relative to $Q$ ($O(Q)$) — (V, 2.16.12)
• osculating hyperplane (vs. tangent non-osculating) — (V, 5.8.7), (V, 5.8.10)

P

• partial degree of an invertible module with respect to the factor $P_i$ — (V, 6.2.6)
• pencils of hyperplane sections — (V, 5.10); see also linear pencil
• positive relative divisor of degree $n$ (${\mathrm{Div}}^n(P/S)$) — (V, 6.2.2)
• positive relative divisor of multidegree $n$ (${\mathrm{Div}}^n(P/S)$, multiprojective case) — (V, 6.2.8)
• prescheme of projective groups (= projective group, = $PGL(E)$) — (V, 6.1.9)
• primary extension of $k$ — (V, 5.3.1.1)
• principal homogeneous fibration with group $O(n{)}_S$ (associated to a smooth quadratic form) — (V, 2.16.12)
• projective base of $P$ (in the linear-systems sense) — (V, 6.4.4)
• projective fibration $P=\mathbb{P}(E)$ — (V, 5.1)
• projective group (= $PGL(E)$) — (V, 6.1.9)
• pseudo-morphism relative to $S$ (= rational map) — (V, 6.3), (V, 6.3.2)

Q

• quadratic form on $E^{\vee }$ (alternative interpretation of $Q\in Sy{m}^2(E)$) — (V, 2.16.1)
• quadric (non-singular, hyperplane sections example) — (V, 5.8.17), (V, 5.9.1)

R

• ramification subprescheme $Ram(V^{\prime }/Y)$ — (V, 1.7)
• rational map relative to $S$ — (V, 6.3)
• regular homomorphism $F_X(-1)\to {\mathcal{O}}_X$ (cuts out a regular sequence) — (V, 5.10.2)
• regular immersion (deduced from a regular homomorphism) — (V, 5.10.2)
• regular morphism — (V, 5.2.8)
• relative divisor over $P/S$ ($\mathrm{Div}(P/S)$) — (V, 5.1.1)
• ruled (for the projective immersion) — (V, 5.8.12) (Grothendieck note), (V, 5.8.13), (V, 5.8.18)

S

• scheme of hyperplanes of $P$ ($P^{\vee }$) — (V, 5.1)
• scheme of zeros of a section $\psi$ of $E$ — (V, 1.5)
• Seidenberg-type theorem (openness of $(S_k)$ / normality loci) — (V, 5.5)
• separable hyperplane section — (V, 5.4.3), (V, 5.8.14)
• singular non-supersingular zero — (V, 2.15.1) (Grothendieck note)
• singular quadratic elementary zero (= elementary singular zero of multiplicity 2) — (V, 2.15.5)
• singular zero of $\varphi$ relative to $Y$ — (V, 1.4)
• singular zero (root) of a section $\varphi$ — (V, 1.1)
• smooth form (on a locally free module of finite type) — (V, 2.15.1)
• smooth quadratic form — (V, 2.15.1), §V.2.16 (chapter title)
• standard quadratic form $Q_n$ on ${\mathbb{Z}}^n$ — (V, 2.16.4)
• sufficiently general hyperplane section — (V, 5.4)
• supersingular point of a locally Noetherian prescheme — (V, 2.15.1) (Grothendieck note)
• supersingular zero of $\varphi$ relative to $Y$ — (V, 1.4)
• supersingular zero (of a section $\varphi$) — (V, 1.1)

T

• tangent hyperplane (to $X$ at $x$) — (V, 5.8.6), (V, 5.8.7)
• tangent sheaf ${\mathcal{G}}_{X/Y}$ (kernel of the augmentation ${\mathcal{P}}_{X/Y}^1\to \mathcal{O}$) — (V, 1.5), footnote $[{}^v-1-3]$
• transversally regular section (of an invertible module) — (V, 6.2.1)

U

• ultra-singular zero $V(\varphi {)}_{ultsing}$ — (V, 2.15.1) (Grothendieck note)
• universal hyperplane $H$ (diagonal section over $P^{\vee }$) — (V, 5.1)

V

• variable hyperplane section — (V, 5.4)
• vector bundle (modern translation of fibré vectoriel $\mathbb{V}(\cdot )$) — (V, 2.16.3), (V, 2.16.11)

W

• without fixed components (linear system) — (V, 6.3)
• without fixed points (family of divisors) — (V, 6.3.1)

Z

• zero set of $\varphi$ ($V(\varphi )$) — (V, 1.1), (V, 1.5)
• zero set of a section $\psi$ ($V(\psi )$) — (V, 1.5)
• zero set singular relative to $Y$ ($V(\varphi {)}_{sing}$) — (V, 1.5), (V, 1.7)
• zero set supersingular relative to $Y$ ($V(\varphi {)}_{supsing}$) — (V, 1.5), (V, 1.7)