Index of notations — EGA V

Source-ordered, with §V section subheadings. Reused notations from EGA I-IV (e.g. ${\mathcal{O}}_X$, ${\Omega }_{X/S}^1$, $\mathrm{Pic}(X)$, Sym, $\Lambda$, $\mathbb{P}$, Proj, Spec, $\to$, $\mapsto$, $\otimes$, $\cong$) are not re-listed here; see $docs/books/ega/iv/index-of-notations.md$.

Note on overloads: the symbol $\mathcal{G}$ denotes the tangent sheaf ${\mathcal{G}}_{X/Y}$ in §V.1 and §V.5.8 (kernel of the augmentation ${\mathcal{P}}_{X/Y}^1\to {\mathcal{O}}_X$), but is repurposed in §V.6.4.3 as the abstract "tangent vector space ${\mathcal{G}}_{w_0}$" of a functor at a chosen point; the same glyph carries two distinct meanings. Similarly the symbol $\mathcal{P}$ denotes the sheaf of principal parts ${\mathcal{P}}_{X/Y}^n$ in §V.1 and a generic projective fibration in §V.5.15; context distinguishes. The notation $(S_n)$ denotes both Serre's depth condition (§V.5.2.8) and a $\mathbb{Z}$-indexed open partition of the base (§V.6.1.1); we attach the subscript range $(S_n{)}_{n\in \mathbb{Z}}$ when ambiguity could arise.

§V.1. Singular and supersingular zeros

• ${\mathfrak{m}}_x$, ${\mathfrak{m}}_x^2$, ${\mathfrak{m}}_x/{\mathfrak{m}}_x^2$ — maximal ideal of ${\mathcal{O}}_{X,x}$, its square, and the Zariski cotangent space. (V, 1.1)
• $t_x$ — dual of ${\mathfrak{m}}_x/{\mathfrak{m}}_x^2$ over $k(x)$ (Zariski tangent space). (V, 1.1)
• $Sy{m}^2({\mathfrak{m}}_x/{\mathfrak{m}}_x^2)$ — module of quadratic forms on $t_x$. (V, 1.1)
• $V(\varphi )$ — zero set / subscheme of zeros of a section $\varphi$. (V, 1.1), (V, 1.5)
• $V(\varphi {)}_{sing}$ — singular zeros of $\varphi$ (relative to $Y$). (V, 1.5)
• $V(\varphi {)}_{supsing}$ — supersingular zeros of $\varphi$ (relative to $Y$). (V, 1.5)
• ${\varphi }_{k^{\prime }}$, $X_{k^{\prime }}$ — base change of $\varphi$ and of $X$ along $k\to k^{\prime }$. (V, 1.3)
• $d^0\varphi =\varphi$ — order-0 principal-parts truncation. (V, 1.5)
• $d_{X/Y}^1\varphi$ — order-1 truncation. (V, 1.5)
• $d_{X/Y}^2\varphi$ — order-2 truncation. (V, 1.5)
• $d_{X/Y}\varphi$ — universal differential of $\varphi$ (restriction of $d^1\varphi$ to $V(\varphi )$). (V, 1.5)
• ${\mathcal{P}}_{X/Y}^n$ — sheaf of principal parts of order $n$ (cf. $(0_{IV},\mathrm{20.4.14})$). (V, 1.5)
• $g{r}^1({\mathcal{P}}_{X/Y}^1)\cong {\Omega }_{X/Y}^1$ — first graded piece. (V, 1.5)
• $g{r}^2({\mathcal{P}}_{X/Y}^2)\cong Sy{m}^2({\Omega }_{X/Y}^1)$ — second graded piece. (V, 1.5)
• $M(\varphi )$ — canonical section of $Sy{m}^2({\Omega }_{X/Y}^1)\otimes {\mathcal{O}}_{V(\varphi {)}_{sing}}$; quadratic form on the cotangent space. (V, 1.5)
• $M(\varphi {)}^{\prime }$ — homomorphism ${\mathcal{G}}_{X/Y}\otimes {\mathcal{O}}_{V(\varphi {)}_{sing}}\to {\Omega }_{X/Y}^1\otimes {\mathcal{O}}_{V(\varphi {)}_{sing}}$ deduced from $M(\varphi )$. (V, 1.5)
• $D(\varphi )=detM(\varphi )$ — discriminant of $M(\varphi )$; section of $\Gamma ({\Omega }_{X/Y}^d{)}^{\otimes 2}\otimes {\mathcal{O}}_{V(\varphi {)}_{sing}}$. (V, 1.5)
• ${\mathcal{G}}_{X/Y}$ — tangent sheaf; kernel of ${\mathcal{P}}_{X/Y}^1\to {\mathcal{O}}_X$. (V, 1.5), footnote $[{}^v-1-3]$
• $V^{\prime }=V(\varphi {)}_{sing}$ — abbreviation in §V.1.7. (V, 1.7)
• $\mu$, $\nu$ — composed homomorphisms in the §V.1.7 diagram. (V, 1.7)
• $Ram(V^{\prime }/Y)$ — ramification subprescheme of $V^{\prime }$ relative to $Y$. (V, 1.7)
• det F — highest exterior power of a locally free module $F$ of finite rank. (V, 1.8)
• ${\Lambda }^{q}Q$, ${\Lambda }^{r}M$ — exterior powers used to detect non-surjectivity loci. (V, 1.7)
• L = det P ⊗ det Q ⊗ det M^{−1} — line bundle in the §V.1.8 lemma. (V, 1.8)

§V.2.15-§V.2.16. Smooth forms; smooth quadratic forms

• $\mathbb{P}(E)$, $P=\mathbb{P}(E)$ — projective fibration. (V, 2.15.1)
• ${\mathcal{O}}_P(n)$ — Serre twist on $P$. (V, 2.15.1)
• $Sy{m}^n(E)$ — $n$-th symmetric power. (V, 2.15.1)
• $V(\varphi )\subset P$ — subscheme of zeros of an $n$-form. (V, 2.15.1)
• $\mathcal{J}$, ${\mathcal{O}}_X/\mathcal{J}$ — ideal and quotient sheaf in an augmentation. (V, 2.15.2)
• ${\mathcal{N}}_{X/Y}=\mathcal{J}/{\mathcal{J}}^2$ — conormal module of $Y=V(\mathcal{J})$ in $X$. (V, 2.15.2)
• $\Psi :Sy{m}_{{\mathcal{O}}_Y}(\mathcal{N})\to g{r}_{\mathcal{J}}({\mathcal{O}}_X)$ — canonical surjection on the associated graded ring. (V, 2.15.2)
• $\mathcal{K}=\ker ({\Psi }_n)$ — kernel in degree $n$ (invertible submodule of $Sy{m}_{{\mathcal{O}}_Y}^n(\mathcal{N})$). (V, 2.15.2)
• $r(A)$ — radical / maximal ideal of a local ring. (V, 2.15.2)
• $d^n\varphi$ — order-$n$ principal part of $\varphi$. (V, 2.15.7)
• $I$ — augmentation ideal of ${\mathcal{P}}_{X^{\prime }/k}^n\otimes k(x)$. (V, 2.15.7)
• $Q\in Sy{m}^2(E)$ — quadratic form. (V, 2.16.1)
• $B$ — symmetric bilinear form on $E^{\vee }$ associated to $Q$. (V, 2.16.1)
• $d(Q)\in \Gamma (det(E{)}^{\otimes 2})$ — ordinary discriminant. (V, 2.16.1)
• $N\subset E^{\vee }$ — "kernel" of the alternating bilinear form in characteristic 2. (V, 2.16.2)
• $Q_n$, $Q_{2m}$, $Q_{2m+1}$ — standard quadratic forms in $n$, 2m, $2m+1$ variables. (V, 2.16.4)
• $\tilde{d}(Q)$ — corrected (adjusted) discriminant of $Q$. (V, 2.16.6), (V, 2.16.7)
• $\mathbb{V}(\cdot )$ — vector-bundle functor associated to a locally free sheaf. (V, 2.16.3), (V, 2.16.11)
• $Isom((E_0,Q_0),(E,Q))$ — functor of isometries. (V, 2.16.11)
• $Isom(E_0,E)$ — functor of isomorphisms of locally free modules. (V, 2.16.11)
• $\mathcal{Q}(E)=\mathbb{V}(Sy{m}^2(E))$ — affine bundle of quadratic forms. (V, 2.16.11)
• $\mathcal{Q}(E)\ast =\mathcal{Q}(E{)}_d$ — open subset of smooth (i.e. $d$-invertible) quadratic forms. (V, 2.16.11)
• $O(n)$ — absolute orthogonal group (stabilizer of $Q_n$ in $\mathrm{Aut}({\mathbb{Z}}^n)$). (V, 2.16.12)
• $O(Q)$ — orthogonal group scheme relative to $Q$. (V, 2.16.12)
• $SO(n)$ — special orthogonal subgroup. (V, 2.16.14)
• ${\mu }_{2,S}$ — group scheme of square roots of unity over $S$. (V, 2.16.14)

§V.5.1. Hyperplane sections — preliminaries and notation

• $P=\mathbb{P}(E)$ — projective fibration. (V, 5.1)
• $P^{\vee }=\mathbb{P}(E^{\vee })$ — scheme of hyperplanes (dual projective fibration). (V, 5.1)
• $L$, L_P, $L_P^{-1}$ — invertible quotient / submodule used to define a hyperplane. (V, 5.1)
• $H_{\xi }$ — hyperplane in $P$ defined by $\xi \in P^{\vee }(S)$. (V, 5.1)
• $H$ — universal / incidence hyperplane in $P{\times }_S{P}^{\vee }$. (V, 5.1)
• $\mathrm{Div}(P/S)$ — functor of relative divisors of $P/S$. (V, 5.1.1)
• $f:X\to P$ — projective immersion / unramified morphism. (V, 5.1)
• $Y_{\xi }=f^{-1}(H_{\xi })$ — hyperplane section associated to $\xi$. (V, 5.1)
• $Y=X{\times }_{P}H$ — total hyperplane section over $P^{\vee }$. (V, 5.1)
• $F$, $G_{\xi }$, $G$ — sheaf on $X$ and its inverse images on $Y_{\xi }$, $Y$. (V, 5.1)

§V.5.2-§V.5.3. Generic hyperplane section

• $\eta$ — generic point of $P^{\vee }$. (V, 5.2)
• $H_{\eta }$, $Y_{\eta }$, $G_{\eta }$ — hyperplane, hyperplane section, sheaf at $\eta$. (V, 5.2)
• $Z_{\eta }$ — inverse image of $Z\subset P$ in $H_{\eta }$. (V, 5.2.2)
• $Y_{i,\eta }$ — irreducible component of $Y_{\eta }$. (V, 5.2.5)
• $copro{f}_{y}G$ — codepth of $G$ at $y$. (V, 5.2.8)
• $(S_k)$, $(R_k)$ — Serre's depth and regularity conditions. (V, 5.2.8), (V, 5.2.11)
• $K$, $L$ — function fields of $X$ and $Y_{\eta }$. (V, 5.3.1)
• $S_1,\cdots ,S_n$ — affine coordinates in $P^{\vee }$. (V, 5.3.1)
• $T_1,\cdots ,T_n$ — affine coordinates in $P$. (V, 5.3.1)
• $t_i\in K$ — image of $T_i$ under $f:X\to P$. (V, 5.3.1)
• $(C_m)$ — non-disconnect-by-codimension-$m$ property. (V, 5.4.4)

§V.5.4-§V.5.5. Variable hyperplane sections; Seidenberg-type theorems

• $E$ — set of $\xi \in P^{\vee }$ such that $Y_{\xi }$ has a given property. (V, 5.4)
• Z_P — set of $\xi \in P^{\vee }$ exceptional for property $P$. (V, 5.8.1)
• ${\varphi }_{\xi }$ — section of ${\mathcal{O}}_X(1)\otimes {\mathcal{O}}_{P^{\vee }}(1)(\xi )$ defining $Y_{\xi }$. (V, 5.5.2)
• $U$, U_P, $V$, V_P — flatness / $P$-regularity loci on $Y$ or on $P^{\vee }$. (V, 5.5.3)-(V, 5.5.7)

§V.5.6-§V.5.8. Connectedness, multisections, dimension of the exceptional set

• $S^{\prime }\to U$ — étale extension. (V, 5.7.1)
• $S^{\prime }$ — multisection of $X$ over $U$. (V, 5.7.1)
• $Z_k$ — exceptional set for the geometric condition $(R_k)$. (V, 5.8.13)
• $Z_n$ — set where $copro{f}_{x}F\ge n$. (V, 5.8.5)
• $Z_0$ — set of $\xi$ with an irreducible component of "dimension too large". (V, 5.8.4)
• $T_i$, $Z_{n,i}$ — irreducible components used in (5.8.4)-(5.8.5). (V, 5.8.4), (V, 5.8.5)
• ${\nu }_{X/P}^{\vee }$, ${\nu }_{X/P}$ — conormal and normal modules of $f:X\to P$. (V, 5.8.6)
• E_P — tautological bundle on $P$. (V, 5.8.6)
• $\psi =(d\varphi ){|}_Y$ — restriction of the differential to $Y$. (V, 5.8.6)
• $Y^{sing}$, $Y^{supsing}$ — singular / supersingular zero loci on $Y$. (V, 5.8.6), (V, 5.8.7)
• $T$ — closure of $Y^{sing}$ in $P^{\vee }$ (reduced induced structure). (V, 5.8.7)
• $H^{\prime }$ — image of the tangent map in $P^{\vee }$. (V, 5.8.7)
• $g:Y^{sing}\to T$ — dominant morphism. (V, 5.8.7)
• $Y^{\prime ,sing}$ — singular locus of $Y^{\prime }=X-T$. (V, 5.8.18)
• $H_x$ — hyperplane in $P^{\vee }$ defined by $x\in P$. (V, 5.8.18)
• $D$ — section of ${\omega }_{X/k}^{\otimes 2}\otimes {\mathcal{O}}_{Y^{sing}}(1,1)$ whose vanishing is $Y^{supsing}$. (V, 5.8.12)

§V.5.9. Change of projective embedding

• ${\mathbb{P}}^{(n)}=\mathbb{P}(Sy{m}^n(\mathcal{E}))$ — $n$-fold Veronese projective fibration. (V, 5.9.1)
• $u_n:\mathbb{P}\to {\mathbb{P}}^{(n)}$ — Veronese immersion. (V, 5.9.1)
• $f_n=u_n\circ f$ — composed unramified morphism. (V, 5.9.2)
• $Y_{\xi }^{(n)}$ — hyperplane section of $X$ relative to $f_n$. (V, 5.9.3)

§V.5.10. Pencils of hyperplane sections

• Y_L — linear pencil of hyperplane sections defined by $L\subset P^{\vee }$. (V, 5.10.1)
• $L_0$ — polar codimension-2 linear subvariety of $\mathbb{P}$ corresponding to $L$. (V, 5.10.2)
• $T=X{\times }_{\mathbb{P}}{L}_0$ — centre of the blow-up associated to a pencil. (V, 5.10.2)
• $F_X(-1)\to {\mathcal{O}}_X$ — regular homomorphism associated to a pencil. (V, 5.10.2)
• $\tilde{X}$ — blow-up of $X$ with centre $T$. (V, 5.10.2), (V, 5.10.3)
• $\mathcal{G}$, $\mathcal{J}$, $\mathcal{K}$, $\mathcal{H}$ — auxiliary modules / ideals in §V.5.10.3. (V, 5.10.3)
• $p=\mathbb{P}(\mathcal{G})$, $p_C$ — projective fibration / conic projection. (V, 5.10.3)
• $I_{k^{\prime }}=\mathrm{Spec}{k}^{\prime }[t]/(t^2)$ — first-order infinitesimal "double point". (V, 5.10.4)

§V.5.11-§V.5.12. Grassmannians; linear sections

• $Gras{s}_n(\mathcal{E})$ — Grassmannian of rank-$n$ locally free quotients of $\mathcal{E}$. (V, 5.11)
• $Grass(\mathcal{E})$ — disjoint sum over all ranks. (V, 5.11)
• $Gras{s}_n(s)$ — subfunctor associated to a decomposition (s) of $\mathcal{E}$. (V, 5.11)
• $Gras{s}^n(\mathbb{P})=Gras{s}_{n+1}(\mathcal{E})$ — Grassmannian of dimension-$n$ linear subvarieties of $\mathbb{P}$. (V, 5.12)
• Grass_n(ℙ) = Grass^{n−1}(ℙ^∨) = Grass_n(ℰ^∨) — Grassmannian of codimension-$n$ linear subvarieties. (V, 5.12)
• $G{r}_m=Gras{s}_m(\mathbb{P})$ — abbreviation in §V.5.12. (V, 5.12)
• $F$ — canonical quotient on $Gras{s}_m$. (V, 5.12)
• $H^{(m)}$ — incidence prescheme for codimension-$m$ linear sections. (V, 5.12)
• $Y^{(m)}=X{\times }_{\mathbb{P}}{H}^{(m)}$; $X^{(m)}$ (Grothendieck's preferred notation) — linear-section total space. (V, 5.12)
• ${\varphi }_m$, ${\varphi }^{(m)}$ — sections defining the linear-section divisor. (V, 5.12)
• $X_{sing}^{(m)}=V_1$, V_0, V_2, $V^{\prime \prime }$, $V^{\prime }-V^{\prime \prime }$ — filtration of the linear-section family. (V, 5.12)
• $V^{(k)}$ — subscheme of $X^{(m)}$ cut out by $\dim T_x\cap L\ge n-m+k$. (V, 5.12)
• $L_{\xi }$ — linear subvariety of codimension $m$ indexed by $\xi \in G{r}_m$. (V, 5.12)

§V.5.14. Conic projections

• $C=\mathbb{P}(F)$ — centre of conic projection. (V, 5.14)
• $Q(C)$ — projective space of codimension-$m$ linear subvarieties of $\mathbb{P}$ containing $C$. (V, 5.14)
• $p_C:\mathbb{P}-C\to \mathbb{P}(G)$ — algebraic conic projection (cf. EGA II). (V, 5.14)
• $p_C^X=p_C\circ f$ — conic projection of $X$ with centre $C$. (V, 5.14)
• $\tilde{X}(C)$, $\tilde{F}(C)$ — extended conic projection space and sheaf. (V, 5.14)
• $C_{\eta }$ — $C$ over the generic point of $Gras{s}_{m+1}$. (V, 5.14)
• $Y_{\eta }$ — scheme-theoretic image of $X_{k(\eta )}$ under $p_{C_{\eta }}$. (V, 5.14.5)

§V.5.15. Axiomatization

• $\mathcal{P}$, $\mathcal{P}$, $G$ — abstract incidence data ($\mathcal{P}$ a projective fibration, $G$ a Grassmannian-type prescheme, $\mathcal{P}$ the incidence prescheme). (V, 5.15)
• $X=X{\times }_{\mathcal{P}}\mathcal{P}$ — abstract analogue of the total hyperplane section. (V, 5.15)
• $N$, $m$ — relative dimensions of $G/S$ and of $\tilde{\mathcal{P}}\to \mathcal{P}$. (V, 5.15)
• $Z$ — inverse image of $Z$ in $X$. (V, 5.15)
• $E(x,V)$ — set of $\xi$ with bad incidence at $(x,V)$. (V, 5.15)

§V.6.1. Invertible sheaves on projective fibrations

• $P=\mathbb{P}(E)$, $f:P\to S$ — projective fibration with structural morphism. (V, 6.1.1)
• $(S_n{)}_{n\in \mathbb{Z}}$ — open decomposition of $S$ indexed by integers. (V, 6.1.1)
• $M\otimes {\mathcal{O}}_P(n)=f^{\ast }(M)(n)$ — twist of an invertible module on a projective fibration. (V, 6.1.1)
• $\mathrm{Pic}(S)$, $\mathrm{Pic}(P)$ — Picard groups. (V, 6.1.3), (V, 6.1.4)
• (*) — canonical homomorphism $\mathrm{Pic}(S)\times \mathbb{Z}\to \mathrm{Pic}(P)$. (V, 6.1.3)
• $\mathbb{Z}(S)$ — locally constant functions $S\to \mathbb{Z}$. (V, 6.1.4)
• $(\ast bis)$ — extension $\mathrm{Pic}(S)\times \mathbb{Z}(S)\to \mathrm{Pic}(P)$. (V, 6.1.4)
• ${\mathrm{Pic}}_{P/S}$ — relative Picard scheme (forward reference). (V, 6.1.4)
• ${\mathrm{Aut}}_S(P)$ — automorphism group functor of $P$ over $S$. (V, 6.1.9)
• $PGL(E)$, $PGL(r)$, $PGL(r{)}_S$ — projective group schemes. (V, 6.1.9)
• $GL(E)$, ${\mathbb{G}}_m$ — linear group scheme and its centre. (V, 6.1.9)

§V.6.2. Relative divisors

• ${\mathrm{Div}}^{+}(P/S)$ — set of positive relative divisors of $P/S$. (V, 6.2.1)
• ${\mathrm{Div}}^n(P/S)$ — degree-$n$ part. (V, 6.2.2)
• ${\mathrm{Div}}_{P/S}^n$ — subfunctor representing ${\mathrm{Div}}^n$. (V, 6.2.3)
• ${\mathrm{Div}}_{P/S}^{+}$ — full divisor functor. (V, 6.2.4)
• ${\mathcal{O}}_i(n)$ — pullback of ${\mathcal{O}}_{P_i}(n)$ to a multiprojective fibration. (V, 6.2.5)
• ${\mathcal{O}}_P(n)$ — multidegree-$n$ twist on a multiprojective fibration. (V, 6.2.5)
• $Sy{m}^n(E)$, $Sy{m}^{n_i}(E_i)$ — symmetric powers. (V, 6.2.2), (V, 6.2.8)
• $Q$ — finitely presented module on $S$ representing pushforwards (cf. (III, §7)). (V, 6.2.10), (V, 6.2.12)
• ${\mathrm{Div}}_{X/S}^L$ — subfunctor parametrizing divisors with ${\mathcal{O}}_X(D)\cong L\otimes f^{\ast }M$. (V, 6.2.10)

§V.6.3. Linear systems and morphisms

• $Z$ — set of fixed points of a family of divisors. (V, 6.3.1)
• $Z^{\prime }$ — set of fixed points in the extended sense. (V, 6.3.1)
• $D$ — family of positive divisors. (V, 6.3.1)
• ᵗD — symmetric image of $D$. (V, 6.3.1)
• ${\mathrm{Div}}_{T/S}$ — divisor functor. (V, 6.3.1)
• $Q=\mathbb{P}(F)=P^{\vee }$ — parametrizing projective fibration. (V, 6.3)
• $\mathbb{P}(n)=\mathbb{P}(Sy{m}^n(F^{\vee }))$ — degree-$n$ divisor space. (V, 6.3)
• $f:X-Z\to P$ — morphism associated to a linear system. (V, 6.3), (V, 6.3.1)
• $H$ — canonical incidence divisor on $P{\times }_S{P}^{\vee }$. (V, 6.3.1)
• $\Phi$, $\mathfrak{U}$ — sets of linear systems / pseudo-morphisms. (V, 6.3.4)
• $U(f)$ — domain of definition of a pseudo-morphism. (V, 6.3.4)
• $L_U=f_U^{\ast }({\mathcal{O}}_P(1))$, $i_{\ast }(L_U)$ — invertible-module construction. (V, 6.3.4)

§V.6.4. Linear systems and invertible modules

• $N$ — invertible module on $X{\times }_S{P}^{\vee }$ corresponding to a divisor $D$. (V, 6.4)
• $u:E\to g_{\ast }(L)$ — datum classifying a linear system. (V, 6.4.1)
• $Q$, $Q\to E^{\vee }$ — module representing $g_{\ast }(L\otimes \cdot )$ and surjection determining a linear system. (V, 6.4.2)
• $Grass(Q)$ — Grassmannian representing classes of linear systems associated to $L$. (V, 6.4.2)
• ${\mathrm{Aut}}_S(P^{\vee })$, ${\mathrm{Aut}}_S(\mathbb{P}(E^{\vee }))$ — automorphism group functor. (V, 6.4.3)
• $Ens(D)$, $\Delta$ — set of divisors arising from a linear system. (V, 6.4.5)
• ${\mathcal{G}}_{w_0}$ — abstract "tangent vector space" of a functor at $w_0$ (re-use of the §V.1 glyph; cf. note above). (V, 6.4.3)
• ${\mathcal{N}}_{D_0/X_0}$ — normal sheaf of D_0 in X_0. (V, 6.4.3)
• $div(\varphi )$ — divisor of a meromorphic function $\varphi$. (V, 6.4.5)
• $E$ (in §V.6.4.5) — $k$-subspace of meromorphic functions defining a linear system. (V, 6.4.5)