The geometric genus of a smooth projective surface is

The irregularity of a smooth projective complex surface is

The Kodaira dimension measures the asymptotic growth of the pluricanonical spaces $H^0(X,K_X^{\otimes m})$. For a surface it takes values $-\infty ,0,1,2$.

An elliptic surface is a smooth projective surface equipped with a surjective morphism to a smooth curve whose generic fiber is a smooth curve of genus one.

A complex K3 surface is a compact, simply connected complex surface with trivial canonical bundle. For projective algebraic surfaces, one can equivalently require a trivial canonical bundle and $H^1(X,{\mathcal{O}}_X)=0$. Nonprojective complex K3 surfaces are also included, which is essential for the full moduli space of Ricci-flat K3 metrics.

An exceptional curve of the first kind is a smooth rational curve $E$ on a smooth surface with $E^2=-1$. It can be contracted to a smooth point.

This is precisely the curve contracted by the Castelnuovo contraction criterion.

Every morphism $f:{\mathbb{P}}^2\to {\mathbb{P}}^n$ satisfies $f^{\ast }\mathcal{O}(1)\cong \mathcal{O}(d)$ for some $d\ge 0$. If $f$ contracts a line, restriction to that line makes $\mathcal{O}(d)$ trivial, hence $d=0$ and all defining sections are constant. Thus the entire morphism is constant.

If $\pi :\tilde{X}\to X$ blows up a smooth point with exceptional curve $E$, thenIf a curve $C$ has multiplicity $m$ at the center, its strict transform is $\tilde{C}={\pi }^{\ast }C-mE$ and ${\tilde{C}}^2=C^2-m^2$.

On a smooth projective surface, two line bundles are represented by divisors $D$ and $E$, and their intersection is the degree of the zero-cycle obtained after moving them into proper position. It is a symmetric bilinear form on the Picard group, and $D\cdot E$ equals the degree of ${\mathcal{O}}_X(D)$ restricted to $E$ when $E$ is an integral curve not contained in $D$.

A semi-additive category has a commutative-monoid enrichment and finite products in a category and coproducts in a category. Composition preserves addition and zero in each variable. Every finite product in a category is canonically a coproduct in a category and conversely, producing finite biproducts. Some authors use "semi-additive structure" for the commutative-monoid enrichment alone; that weaker convention also permits one-object categories without finite products in a category.

A preadditive category instead has abelian-group-valued hom sets. Semi-additivity alone does not require additive inverses.

An additive functor between additive categories induces a group homomorphism on every hom-group. It therefore preserves zero morphisms and finite biproducts.

A total-spin-$J$ sector is an irreducible subspace on which ${\mathbf{J}}^2$ has eigenvalue $J(J+1){\hslash }^2$. For two spin-one particles, Clebsch-Gordan decomposition gives sectors $J=0,1,2$ of dimensions one, three, and five.

An achronal boundary is the boundary of a chronological future or past. Its null generators cease to belong to the boundary after developing a conjugate point or intersecting another generator in the relevant way.

A hyperaccreting black-hole disk has an accretion rate so large that photon cooling can be trapped and neutrino emission can dominate the cooling of its hot dense inner region. Such disks are models for the central engines of some gamma-ray bursts.

An accretion flow carries matter inward through a gravitationally bound system. In a disk, angular-momentum transport permits the inward drift while stresses carry angular momentum outward.

The Z4 formulation introduces a spacetime vector $Z_{\mu }$ into Einstein's equations so that the Hamiltonian and momentum constraints become part of a dynamical system. Its normal projection is commonly written $\Theta =-n^{\mu }{Z}_{\mu }$.

For a unit timelike normal $n^{\mu }{n}_{\mu }=-1$, the spatial projection tensor annihilates $n^{\mu }$ and projects every spacetime tensor onto a spatial hypersurface.

For a separated duct mode with local width $h(X)$ and nonzero axial wavenumber $k(X)$, conservation of modal flux gives amplitude proportional to $[h(X)k(X){]}^{-1/2}$. The phase is ${ϵ}^{-1}{\int }^{X}k(\xi )d\xi$.

A matter wave is the quantum wave associated with a material particle. Its phase can interfere after coherent paths recombine.

In the position representation, a matter wave is described by a wavefunction: a complex number probability amplitude, with detection probabilities given by the Born rule. A localized wave packet combines different momenta; its envelope can move while its individual quantum phase crests move at another speed. Neither curve is a literal mechanical oscillation of material along a particle's path. The de Broglie phase harmony explains the historical route to $\lambda =h/p$; the relativistic matter-wave velocities distinguish the two speeds. The Davisson–Germer experiment and the Thomson electron transmission diffraction experiment connect that spatial quantum phase to measured electron diffraction.

A discharge coefficient multiplies an ideal inviscid opening-flow rate to represent contraction and dissipative losses. Its value depends on the opening geometry and flow regime.