Fano Threefold Weighted Complete Intersections

• Fano threefold weighted complete intersections are three-dimensional projective varieties defined by weighted homogeneous polynomials, offering concrete examples of Fano varieties.
• They feature explicit numerical classifications, birational rigidity phenomena, and rich moduli and toric interpretations in modern algebraic geometry.
• Applications include precise computation of Hodge invariants, higher Chern character positivity conditions, and innovative constructions via key varieties and unprojection techniques.

Searching arXiv for recent and foundational papers on Fano threefold weighted complete intersections. Fano threefold weighted complete intersections are three-dimensional projective varieties with ample anticanonical divisor that are realized as complete intersections of weighted homogeneous equations in a weighted projective space, or, in the singular Mori-theoretic setting, as quasismooth well-formed weighted complete intersections with terminal singularities and Picard rank $1$. They form one of the most explicit and extensively studied classes of Fano threefolds: their ambient grading makes the canonical class, singularities, Hodge theory, higher Chern-character positivity, Sarkisov links, and moduli constructions unusually concrete (Guerreiro et al., 18 Aug 2025).

1. Definitions and ambient geometry

A weighted projective space is

$$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$

A weighted complete intersection $X\subset \mathbb{P}(a_0,\dots,a_N)$ of multidegree $(d_1,\dots,d_k)$ is the common zero locus of $k$ weighted homogeneous polynomials of degrees $d_1,\dots,d_k$, with codimension $k$. The ambient space is well formed if

$\gcd(a_0,\dots,\widehat{a_i},\dots,a_N)=1$

for all $i$, and $X$ is well formed if the ambient is well formed and

$$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$0

A weighted complete intersection is not an intersection with a linear cone if $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$1 for all $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$2 (Vikulova, 2022).

In the smooth setting, a Fano variety is a smooth complex projective variety $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$3 with ample anticanonical divisor $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$4, equivalently $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$5 ample. In the Mori-theoretic setting used in the birational literature, a $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$6-Fano threefold is a normal projective threefold with terminal singularities, $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$7-factoriality, Picard number $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$8, and ample $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$9 (Okada, 2014). For a codimension-$X\subset \mathbb{P}(a_0,\dots,a_N)$0 weighted complete intersection

$X\subset \mathbb{P}(a_0,\dots,a_N)$1

adjunction gives

$X\subset \mathbb{P}(a_0,\dots,a_N)$2

and similarly, for a general weighted complete intersection,

$X\subset \mathbb{P}(a_0,\dots,a_N)$3

Thus the Fano condition is

$X\subset \mathbb{P}(a_0,\dots,a_N)$4

(Vikulova, 2022).

Quasismoothness is formulated on the affine cone. If $X\subset \mathbb{P}(a_0,\dots,a_N)$5 is defined by a weighted homogeneous ideal $X\subset \mathbb{P}(a_0,\dots,a_N)$6, its affine cone $X\subset \mathbb{P}(a_0,\dots,a_N)$7 is smooth away from the vertex when $X\subset \mathbb{P}(a_0,\dots,a_N)$8 is quasismooth. In the threefold literature, quasismooth and well-formed weighted complete intersections are the standard explicit models for terminal $X\subset \mathbb{P}(a_0,\dots,a_N)$9-Fano threefolds (Okada, 31 May 2025).

2. Classification landscape

The basic numerical classification of Fano threefold weighted complete intersections is finite and highly explicit. The principal families discussed in the literature are summarized below.

Class	Count	Description
Index-$(d_1,\dots,d_k)$0 weighted hypersurfaces	95	Quasi-smooth Fano threefold hypersurfaces (Guerreiro et al., 18 Aug 2025)
Higher-index weighted hypersurfaces	35	Quasi-smooth Fano threefold hypersurfaces (Guerreiro et al., 18 Aug 2025)
Codimension-$(d_1,\dots,d_k)$1, index-$(d_1,\dots,d_k)$2 WCIs	85	Anticanonically embedded $(d_1,\dots,d_k)$3-Fano 3-fold WCIs (Okada, 2013)
Codimension-$(d_1,\dots,d_k)$4, higher index WCIs	40	Families numbered $(d_1,\dots,d_k)$5 in GRDB-based classification (Guerreiro, 2023)
Codimension $(d_1,\dots,d_k)$6 WCI	1	Complete intersection of three quadrics in $(d_1,\dots,d_k)$7 (Guerreiro et al., 18 Aug 2025)

For codimension $(d_1,\dots,d_k)$8, Iano-Fletcher’s list contains $(d_1,\dots,d_k)$9 anticanonically embedded $k$0-Fano $k$1-fold weighted complete intersections; this is the class analyzed in detail by Okada’s birational series (Okada, 2013). In the higher-index codimension-$k$2 case, the GRDB-based analysis identifies $k$3 families with Fano index at least $k$4 (Guerreiro, 2023).

In the smooth well-formed setting, weighted complete intersections of dimension $k$5 are very restricted. Their Hodge-theoretic behavior shows that every Fano threefold weighted complete intersection is of curve type, while the only $k$6-homologically minimal and diagonal example is the smooth quadric threefold (Przyjalkowski et al., 2018). This places smooth threefold weighted complete intersections close to the classical rank-$k$7 Fano threefolds arising as complete intersections of quadrics and cubics.

A further extension replaces ordinary weighted projective spaces by fake weighted projective spaces. In that toric framework, terminal or Gorenstein Fano threefold complete intersections are classified by Cox-ring degrees, relation degrees, and torsion in the class group, and many of the resulting families descend to ordinary weighted projective models by “downgrading” the torsion data (Hausen et al., 2020).

3. Higher Fano structures and positivity of Chern characters

Beyond the usual Fano condition, one can impose positivity on higher Chern characters. If $k$8 denotes the $k$9-th Chern character of $d_1,\dots,d_k$0, then $d_1,\dots,d_k$1 is called positive if

$d_1,\dots,d_k$2

for every effective $d_1,\dots,d_k$3-cycle $d_1,\dots,d_k$4, and a smooth Fano variety is $d_1,\dots,d_k$5-Fano if $d_1,\dots,d_k$6 is positive for all $d_1,\dots,d_k$7 (Vikulova, 2022).

For a smooth well formed weighted complete intersection

$d_1,\dots,d_k$8

of multidegree $d_1,\dots,d_k$9, the Chern characters take the form

$k$0

Hence the $k$1-Fano condition becomes

$k$2

Moreover, for smooth well formed Fano weighted complete intersections that are not intersections with a linear cone, it suffices to check positivity at a single value: $k$3 (Vikulova, 2022).

In dimension $k$4, the general bound

$k$5

specializes to

$k$6

Thus a smooth Fano threefold weighted complete intersection can never be $k$7-Fano for $k$8. The extremal case $k$9 is completely rigid: if a smooth Fano threefold weighted complete intersection is $\gcd(a_0,\dots,\widehat{a_i},\dots,a_N)=1$0-Fano, then the ambient weighted projective space must be ordinary projective space, and the threefold must be a smooth complete intersection of quadrics in $\gcd(a_0,\dots,\widehat{a_i},\dots,a_N)=1$1. In particular, genuinely weighted threefold examples cannot be $\gcd(a_0,\dots,\widehat{a_i},\dots,a_N)=1$2-Fano (Vikulova, 2022).

This higher-positivity result isolates the classical complete intersections of quadrics as the only $\gcd(a_0,\dots,\widehat{a_i},\dots,a_N)=1$3-Fano threefold weighted complete intersections and shows that higher Chern-character positivity is far more restrictive than the ordinary Fano condition.

4. Hodge theory, Torelli phenomena, and coregularity

The Hodge theory of weighted complete intersections is computable through a bigraded Jacobian ring. For a quasi-smooth weighted complete intersection

$\gcd(a_0,\dots,\widehat{a_i},\dots,a_N)=1$4

one sets

$\gcd(a_0,\dots,\widehat{a_i},\dots,a_N)=1$5

in a bigraded polynomial ring and defines the Jacobi ring

$\gcd(a_0,\dots,\widehat{a_i},\dots,a_N)=1$6

If $\gcd(a_0,\dots,\widehat{a_i},\dots,a_N)=1$7, then for $\gcd(a_0,\dots,\widehat{a_i},\dots,a_N)=1$8 and $\gcd(a_0,\dots,\widehat{a_i},\dots,a_N)=1$9,

$i$0

and the infinitesimal Torelli map is identified with multiplication in $i$1 (Licht, 2022).

A central threefold application concerns hyperelliptic Fano threefolds of Picard rank $i$2, index $i$3, and degree $i$4. Every such threefold is a weighted complete intersection

$i$5

with $i$6, $i$7, and $i$8 a smooth quadric. The hyperelliptic involution acts by $i$9, and the $X$0-invariant part of the infinitesimal Torelli map is injective. The same Jacobi-ring analysis also shows that $X$1 acts faithfully on $X$2, while the kernel of the action on $X$3 is generated by the hyperelliptic involution (Licht, 2022).

For smooth well-formed Fano weighted complete intersections, Hodge level is also explicit. If $X$4 is not an odd-dimensional quadric, then

$X$5

where $X$6 is defined from the index and maximal degree of the complete intersection. In dimension $X$7, every Fano threefold weighted complete intersection is of curve type, and the only diagonal and $X$8-homologically minimal case is the quadric threefold (Przyjalkowski et al., 2018).

A different degeneration invariant is coregularity, defined via dual complexes of Calabi–Yau pairs. Explicit examples show that the classical complete-intersection-type Fano threefolds of degrees $X$9, $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$00, $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$01, and $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$02 admit members of coregularity zero. In particular, the weighted sextic double solid

$$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$03

is a smooth Fano threefold of family $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$04 with coregularity zero, and together with analogous quartic, $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$05, and $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$06 examples this implies that every family of smooth Fano threefolds contains an element of coregularity zero (Zhakupov, 2024).

5. Birational geometry, rigidity, and Mori fibre structures

The birational theory of Fano threefold weighted complete intersections is organized by the Sarkisov program. A Mori fibre space is a projective $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$07-factorial terminal variety $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$08 with $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$09 relatively ample, $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$10, and $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$11. A Fano threefold of Picard rank $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$12 is itself a Mori fibre space over a point. Birational rigidity means that every birational map to a Mori fibre space is square birational to the original model; birational solidity means that no birational map exists to a Mori fibre space over a positive-dimensional base (Okada, 2014).

For the $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$13 codimension-$$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$14 anticanonically embedded $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$15-Fano $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$16-fold WCIs, Okada determined maximal centers and constructed explicit Sarkisov links or birational involutions. The result is a sharp dichotomy: $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$17 families are birationally rigid and the remaining $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$18 are birationally nonrigid (Okada, 2013). In the continuation of that program, among the remaining flexible families with birational counterpart $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$19 carrying a $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$20 or $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$21 point, $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$22 families are shown to be birationally birigid: a general member has exactly two Mori fibre structures, namely the original codimension-$$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$23 weighted complete intersection and its weighted hypersurface counterpart (Okada, 2014).

Weighted complete intersections also furnish the first examples of $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$24-Fano threefolds with exactly three birational Mori fibre structures. In the construction

$$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$25

of degree $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$26, together with complete intersections

$$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$27

of type $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$28, one obtains three birational Mori fibre structures $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$29 in the asymmetric case, and two in the symmetric case. This also shows that the number of birational Mori fibre structures is neither upper nor lower semicontinuous in families (Okada, 2014).

The higher-index codimension-$$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$30 families are systematically non-rigid. For quasismooth Fano threefold complete intersections appearing in the GRDB as codimension-$$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$31 complete intersections, linear cyclic quotient singularities are introduced and proved to be maximal centers. Each such threefold has a linear cyclic quotient singularity leading to a Sarkisov link, and as a consequence, if a Fano threefold weighted complete intersection is birationally rigid, then its Fano index is $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$32. When the target is a strict Mori fibre space, it is explicitly a del Pezzo fibration of degrees $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$33, $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$34, or $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$35, or a conic bundle over a weighted projective plane with at most $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$36 singularities (Guerreiro, 2023).

A model higher-index solidity result is now known. Any quasismooth Fano threefold weighted complete intersection of type

$$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$37

is birationally solid. It is birational to a degree-$$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$38 weighted hypersurface

$$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$39

with a $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$40 point, but to no Mori fibre space over a positive-dimensional base. This is described as the first example of a birationally solid Fano $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$41-fold weighted complete intersection of codimension $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$42 and index $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$43 (Okada, 31 May 2025).

6. Prime $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$44-Fano threefolds of anticanonical codimension $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$45 and key varieties

A major extension of the weighted-complete-intersection paradigm replaces ordinary weighted projective ambient spaces by weighted projectivizations of specially constructed key varieties. In this framework, a prime $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$46-Fano threefold is a normal projective threefold with terminal singularities and ample $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$47, such that the anticanonical divisor generates numerical divisor classes. The anticanonical codimension $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$48 is the codimension of the anticanonical embedding associated to the graded ring

$$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$49

(Takagi, 2024).

Takagi’s constructions use affine key varieties $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$50 and $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$51, together with weighted projectivizations $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$52, $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$53, $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$54, and the weighted cone $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$55. These varieties arise from explicit unprojection constructions and carry $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$56-fibration structures on suitable partial projectivizations. Weighted complete intersections inside them produce prime $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$57-Fano threefolds of anticanonical codimension $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$58 (Takagi, 2021).

The 2024 construction realizes $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$59 GRDB classes using weighted projectivizations of $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$60 and $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$61 classes using weighted projectivizations of $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$62 or its cone. Together with earlier constructions of Coughlan–Ducat and Takagi, prime $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$63-Fano $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$64-folds of anticanonical codimension $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$65 are constructed for $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$66 classes among the $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$67 classes in the GRDB (Takagi, 2024). For these threefolds, the ambient weighted projectivizations and cut-out equations are chosen so that the Hilbert numerator, ambient weights, and basket of singularities agree exactly with the GRDB entry, while a general anticanonical divisor is a quasi-smooth K3 surface with only Du Val singularities of type $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$68 (Takagi, 2021).

This codimension-$$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$69 theory shows that “weighted complete intersection” in the modern Fano-threefold literature is broader than complete intersections in a single weighted projective space: key-variety formats, unprojection, and weighted cones provide a unified way to construct almost all known prime $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$70-Fano threefolds of anticanonical codimension $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$71.

7. Toric and moduli perspectives

Non-degenerate toric complete intersections supply another structured class of Fano threefold weighted complete intersections. In a toric variety $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$72, a non-degenerate complete intersection is obtained from a system of Laurent polynomials whose face systems are all smooth of the expected codimension. For such a complete intersection $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$73, the anticanonical complex $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$74 generalizes the Fano polytope and controls discrepancies combinatorially: $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$75 for exceptional divisors corresponding to rays in a toric resolution. Terminality and canonicality become lattice-point conditions inside $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$76 (Hausen et al., 2020).

This method yields a classification of non-toric terminal Fano general complete intersection threefolds in fake weighted projective spaces, and it interfaces directly with Cox-ring descriptions of weighted complete intersections (Hausen et al., 2020). In the Gorenstein case, the later classification of general toric complete intersection threefolds in fake weighted projective spaces produces exactly $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$77 $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$78-factorial Gorenstein Fano families of Picard rank $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$79: $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$80 hypersurfaces, $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$81 codimension-$$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$82 complete intersections, and $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$83 codimension-$$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$84 complete intersections (Hausen et al., 13 Oct 2025).

Moduli theory adds a complementary viewpoint. For log pairs formed by a complete intersection of two quadrics and a hyperplane, the K-moduli compactification is identified with a VGIT quotient, and the first wall crossing occurs at

$$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$85

for the corresponding log K-moduli problem (Papazachariou, 2022). In dimension $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$86, the same paper explicitly describes the K-moduli of the Mori–Mukai family $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$87, whose members can be viewed as blow ups of complete intersections of two quadrics in dimension three, and proves

$$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$88

(Papazachariou, 2022).

Taken together, the toric, GIT, and K-moduli viewpoints show that Fano threefold weighted complete intersections are not only explicit projective models but also tractable objects in discrepancy theory, variation of geometric invariant theory, and moduli compactification. The subject now spans smooth higher-Fano positivity, singular $$\mathbb{P}(a_0,\dots,a_N)=\operatorname{Proj}\,\mathbb{C}[x_0,\dots,x_N], \qquad \deg x_i=a_i.$$89-Fano birational geometry, key-variety constructions, toric and fake-weighted classifications, and explicit K-moduli descriptions (Guerreiro et al., 18 Aug 2025).