Proof. Notice that the syzygy of each such a curve C is of the form

$$ \begin{align*}0\ \longrightarrow \ \mathcal{O}_{{\mathbb P}^3}(-4)^{\oplus2}\ \longrightarrow \ \mathcal{O}_{{\mathbb P}^3}(-3)^{\oplus2}\oplus \mathcal{O}_{{\mathbb P}^3}(-2)\ \longrightarrow \ \mathscr{I}_{C} \ \longrightarrow \ 0.\end{align*} $$

Then one can argue as in [Reference Liu and ZhaoLZ25, Lemma 4.17].

Proof. Notice that there exists an exact sequence

$$ \begin{align*}0\longrightarrow \mathcal{O}_X(L-S)\longrightarrow \mathcal{O}_X(L)\longrightarrow \mathcal{O}_S(L)\longrightarrow 0,\end{align*} $$

where $L-S\sim -H$ is antieffective. By Kawamata-Viehweg vanishing theorem, we have that

$$ \begin{align*}h^0(X,\mathcal{O}_X(-H))=h^1(X,\mathcal{O}_X(-H))=0.\end{align*} $$

Then the desired isomorphism follows immediately. The dimension of the vector spaces is ${\frac {1}{2}(L|_S)^2+2=6}$ . To prove the second statement, it suffices to observe that a general element $S\in |-K_X|$ is identified with a general quartic K3 surface in $\mathbb {P}^3$ containing a smooth rational normal curve $\Gamma $ .

Proof. The proof follows by the same reasoning as in [Reference Liu and ZhaoLZ25, Proposition 3.6].

Proof. As $(S,L)$ is unigonal, then one has that $L=E+5F$ , where E is a rational curve and F is the class of a genus one curve. The linear series $|F|$ induces a elliptic fibration $f:S\rightarrow \mathbb {P}^1$ , and E is an f-section. It is easy to see that $\ell $ is contained in a (reducible) fiber of f, denoted by $F_0$ . Writing $F_0=\ell +F_0'$ , one has that

$$ \begin{align*}(F_0'.L)\ = \ (F_0-\ell.L)\ =\ 1-1 \ = \ 0,\end{align*} $$

which contradicts with the ampleness of L.

Proof. We will show that the linear series $|2\mu ^{*}L-E|$ does not have base component which intersects negatively with $2\mu ^{*}L-E$ . By Lemma 3.7, the polarized K3 surface $(S,L)$ is either of generic case or hyperelliptic case. In the former case, S is embedded by $|L|$ into ${\mathbb P}^5$ . It is easy to see that $\mathcal {I}_{\ell /S}(2L)$ is globally generated since $\mathcal {I}_{\ell /{\mathbb P}^5}(2)$ is, and hence any base component of $|2\mu ^{*}L-E|$ (if exists) is $\mu $ -exceptional, on which $2\mu ^{*}L-E$ is positive.

If $(S,L)$ is hyperelliptic, let $f:S\rightarrow R\subseteq \mathbb {P}^5$ be the double cover induced by the linear series $|L|$ . The image of $\ell $ , denoted by $\ell _0$ , is a (reduced) line in $\mathbb {P}^5$ as

$$ \begin{align*}(\ell_0.\mathcal{O}_{\mathbb{P}^5}(1)) \ =\ (f_{*}\ell.\mathcal{O}_{\mathbb{P}^5}(1)) \ =\ (\ell.f^{*}\mathcal{O}_{\mathbb{P}^5}(1))\ =\ (\ell.L)\ = \ 1.\end{align*} $$

Since $f:S\rightarrow R$ is a double cover, then there is another line $\ell '$ (not necessarily distinct from $\ell $ ) on X such that $f^{*}\ell _0=\ell +\ell '$ . Since $\mathcal {I}_{\ell _0/{\mathbb P}^5}(2)$ is globally generated, then the only possible non- $\mu $ -exceptional base component of $|2\mu ^*L-E|$ is the strict transform $\widetilde {\ell }'$ of $\ell '$ . However, one has that

$$ \begin{align*}(2\mu^{*}L-E.\widetilde{\ell}') \ \geq \ (2L.l')-(l.l')\ \geq\ 2-1 \ = \ 1.\end{align*} $$

On the other hand, notice that $2\mu ^{*}L-E$ is $\mu $ -ample, and hence $2\mu ^{*}L-Q$ is nef.

Proof. The same argument as in [Reference Liu and ZhaoLZ25, Lemma 3.8] applies.

Proof. Consider the exact sequence

$$ \begin{align*}0\longrightarrow \mathcal{O}_{\mathscr{X}}(-K_{\mathscr{X}}-\mathscr{X}_0)\longrightarrow \mathcal{O}_{\mathscr{X}}(-K_{\mathscr{X}})\longrightarrow \mathcal{O}_{\mathscr{X}_0}(-K_{\mathscr{X}_0})\longrightarrow 0.\end{align*} $$

Applying $\pi _{*}$ , one obtains by the Kawamata-Viehweg vanishing theorem that the restriction

$$ \begin{align*}H^0(\mathscr{X},\mathcal{O}_{\mathscr{X}}(-K_{\mathscr{X}}))\longrightarrow H^0(\mathscr{X}_0,\mathcal{O}_{\mathscr{X}_0}(-K_{\mathscr{X}_0}))\end{align*} $$

is surjective. Moreover, if S is a general member in $|-K_{\mathscr {X}_0}|$ , then one can choose $\mathscr {S}$ to be general, and hence $\mathscr {S}_t\in |-K_{\mathscr {X}_t}|$ is general for any general point $t\in T$ .

Proof. For any $0<\epsilon \ll 1$ , since $\widetilde {\mathscr {Q}}$ is f-ample, then we know that $2\widetilde {\mathcal {L}}-(1-\epsilon )\widetilde {\mathscr {Q}}$ is $\widetilde {\pi }$ -ample, and hence $(2\widetilde {L}-(1-\epsilon )\widetilde {Q})|_{\widetilde {S}}$ is ample on the K3 surface $\widetilde {S}$ .

We claim that $\widetilde {S}$ is represented by a point in $\mathscr {F}_{\Lambda _2}$ (cf. Section 3.2). Indeed, by deforming to a family of K3 surface in $|-K_{\mathscr {X}_t}|$ (cf. Lemma 4.4), one sees that the divisors $\widetilde {H}|_{\widetilde {S}}\sim (\widetilde {L}-\widetilde {Q})|_{\widetilde {S}}$ and $\widetilde {E}|_{\widetilde {S}}\sim (2\widetilde {L}-3\widetilde {Q})|_{\widetilde {S}}$ generate a primitive sublattice of $\operatorname {\mathrm {Pic}}(\widetilde {S})$ which is isometric to $\Lambda _2$ where $(2\widetilde {L}-(1-\epsilon )\widetilde {Q})|_{\widetilde {S}}$ corresponds to the vector $h_2$ .

By Lemma 3.9, we know that $\widetilde {S}$ is represented by a point in $\mathscr {F}_{\Lambda _1}$ as well. In particular, $(\widetilde {L}-\epsilon \widetilde {Q})|_{\widetilde {S}}$ is also ample for any $0<\epsilon \ll 1$ , and hence $\widetilde {L}|_{\widetilde {S}}$ is nef. The degree of $\widetilde {L}|_{\widetilde {S}}$ is invariant under deformation, as $\widetilde {\mathcal {L}}$ is a Cartier divisor on $\widetilde {\mathscr {X}}$ .

Proof. Let $\widetilde {S}\in |-K_{\widetilde {X}}|$ be a K3 surface. By Lemma 4.5 we see that $(\widetilde {S},\widetilde {L}|_{\widetilde {S}})$ is a quasi-polarized degree $8$ K3 surface, and hence

$$ \begin{align*}h^0(\widetilde{S},\widetilde{L}|_{\widetilde{S}})\ =\ \frac{1}{2}({\widetilde{L}}|_{\widetilde{S}})^2+2\ =\ 6.\end{align*} $$

Since $\widetilde {S}\sim -K_{\widetilde {X}}$ is Cartier, we have a short exact sequence

$$\begin{align*}0 \to \mathcal{O}_{\widetilde{X}}({\widetilde{L}}- \widetilde{S}) \to \mathcal{O}_{\widetilde{X}}({\widetilde{L}}) \to \mathcal{O}_{\widetilde{S}}({\widetilde{L}}|_{\widetilde{S}})\to 0. \end{align*}$$

Notice that ${\widetilde {L}}-\widetilde {S}\sim \widetilde {L}+K_{\widetilde {X}}\sim -\widetilde {H}$ is not effective. Hence by taking the long exact sequence we see that $H^0(\widetilde {X},\mathcal {O}_{\widetilde {X}}({\widetilde {L}}))\hookrightarrow H^0(\widetilde {S},{\widetilde {L}}|_{\widetilde {S}})$ is injective, and thus $h^0(\widetilde {X},\mathcal {O}_{\widetilde {X}}({\widetilde {L}}))\leq 6$ . On the other hand, by upper semi-continuity, we have that

$$ \begin{align*}h^0(\widetilde{X},\mathcal{O}_{\widetilde{X}}({\widetilde{L}}))\ \geq\ h^0(\widetilde{\mathscr{X}}_t,\mathcal{O}_{\widetilde{\mathscr{X}}_t}({\widetilde{\mathcal{L}}_t}))\ =\ 6\end{align*} $$

for a general $t\in T$ . Therefore, one has $h^0(\widetilde {X},\mathcal {O}_{\widetilde {X}}({\widetilde {L}}))=6$ , and the restriction map is an isomorphism.

Proof. It suffices to show that $h^0(\widetilde {X},\mathcal {O}_{\widetilde {X}}(\widetilde {Q}))\leq 1$ as $\widetilde {Q}$ is effective. For a general K3 surface $\widetilde {S}\in |-K_{\widetilde {X}}|$ , $\widetilde {L}|_{\widetilde {S}}$ is big and nef. Moreover, by taking the ample model, denoted by $\overline {S}$ , of $\widetilde {S}$ with respect to $\widetilde {L}|_{\widetilde {S}}$ , one get a birational morphism which sends $\widetilde {Q}|_{\widetilde {S}}$ to a line l. Thus

$$ \begin{align*}h^0(\widetilde{S},\mathcal{O}_{\widetilde{S}}(\widetilde{Q}))\ = \ h^0(\overline{S},\mathcal{O}_{\overline{S}}(l))\ =\ 1.\end{align*} $$

As $\widetilde {Q}-\widetilde {S}\sim -2\widetilde {H}$ is not effective, one has an injection $H^0(\widetilde {X},\mathcal {O}_{\widetilde {X}}(\widetilde {Q}))\hookrightarrow H^0(\widetilde {S},\mathcal {O}_{\widetilde {S}}(\widetilde {Q}))$ , which proves the statement.

Proof. Let $\widetilde {S}\in |-K_{\widetilde {X}}|$ be a K3 surface. By Lemma 4.5 we see that $(\widetilde {S},{\widetilde {L}}|_{\widetilde {S}})$ is a quasi-polarized K3 surface of degree $6$ , and hence

$$ \begin{align*}h^0(\widetilde{S},2{\widetilde{L}}|_{\widetilde{S}})\ =\ \frac{1}{2}(2{\widetilde{L}}|_{\widetilde{S}})^2+2\ =\ 18.\end{align*} $$

Arguing as in the proof of Lemma 4.6, we have a long exact sequence

$$\begin{align*}0 \to H^0(\widetilde{X}, \mathcal{O}_{\widetilde{X}}(2{\widetilde{L}} - \widetilde{S})) \to H^0(\widetilde{X}, \mathcal{O}_{\widetilde{X}}(2{\widetilde{L}})) \to H^0(\widetilde{S}, \mathcal{O}_{\widetilde{S}}(2{\widetilde{L}|_{\widetilde{S}}})). \end{align*}$$

Notice that $2{\widetilde {L}}-\widetilde {S}\sim 2{\widetilde {L}}+K_{\widetilde {X}}\sim \widetilde {Q}$ and that $h^0(\widetilde {X},\mathcal {O}_{\widetilde {X}}(\widetilde {Q}))=1$ (cf. Lemma 4.7). Hence the above long exact sequence implies that

$$ \begin{align*}h^0(\widetilde{X},\mathcal{O}_{\widetilde{X}}(2{\widetilde{L}}))\ \leq\ h^0(\widetilde{S},2{\widetilde{L}}|_{\widetilde{S}})+1\ =\ 19.\end{align*} $$

On the other hand, by upper semi-continuity, we have that

$$ \begin{align*}h^0(\widetilde{X},\mathcal{O}_{\widetilde{X}}(2{\widetilde{L}}))\ \geq\ h^0(\widetilde{\mathscr{X}}_t,\mathcal{O}_{\widetilde{\mathscr{X}}_t}(2{\widetilde{\mathcal{L}}_t}))\ =\ 19\end{align*} $$

for any general $t\in T$ . Thus the proof is finished.

Proof. We first prove that $\widetilde {L}$ is a nef divisor. Recall that we have $2\widetilde {L}\sim -K_{\widetilde {X}}+\widetilde {Q}$ . As $-K_{\widetilde {X}}$ is big and nef, and $\widetilde {Q}$ is g-ample, then we have that $(\widetilde {L}.C)>0$ for any g-exceptional curve $C\subseteq \widetilde {X}$ .

We claim that the base locus of the linear series $|2\widetilde {L}|$ is either some isolated points, or is contained in the g-exceptional locus, so $\widetilde {L}$ is nef. By Lemma 4.5, we know that $|2\widetilde {L}|_{\widetilde {S}}|$ is base-point-free, where $\widetilde {S}\in |-K_{\widetilde {X}}|$ is a general member. Suppose that $\widetilde {C}\subseteq \operatorname {\mathrm {Bs}}|2\widetilde {L}|$ is a curve which is not contracted by g. Then the intersection $\widetilde {C}\cap \widetilde {S}$ is nonempty and consists of finitely many points, which are all base points of $|2\widetilde {L}|_{\widetilde {S}}|$ (cf. Lemma 4.8). This leads to a contradiction.

Since $\widetilde {L}= \widetilde {\mathcal {L}}|_{\widetilde {\mathscr {X}}_0}$ is nef, and $\widetilde {\mathcal {L}}|_{\widetilde {\mathscr {X}}_t}$ is nef for any $t\in T\setminus \{0\}$ as $\widetilde {\mathscr {X}}_t\simeq \mathscr {X}_t$ is a smooth Fano threefold in the family 2.19, we conclude that $\widetilde {\mathcal {L}}$ is $\widetilde {\pi }$ -nef. This implies the $\widetilde {\pi }$ -semiampleness of $\widetilde {\mathcal {L}}$ by Kawamata–Shokurov base-point-free theorem, as $\widetilde {\mathscr {X}}$ is of Fano type over T. Since $\widetilde {\mathcal {L}}|_{\widetilde {\mathscr {X}}_t}$ is big for a general t, we get the $\widetilde {\pi }$ -bigness of $\widetilde {\mathcal {L}}$ .

Proof. We first show that V is normal and $\phi _0$ is birational. Since both $\widetilde {\mathcal {L}}$ and $-K_{\widetilde {\mathscr {X}}}$ are nef and big over T and $\widetilde {\mathscr {X}}$ is klt, then Kawamata–Viehweg vanishing theorem implies that

$$ \begin{align*}R^i \widetilde{\pi}_* \widetilde{\mathcal{L}}^{\otimes m} \ =\ 0\end{align*} $$

for any $i> 0$ and $m\in {\mathbb N}$ . Thus, by cohomology and base change, the sheaf $\widetilde {\pi }_* \widetilde {\mathcal {L}}^{\otimes m}$ is locally free and satisfies that

$$ \begin{align*}\left(\widetilde{\pi}_* \widetilde{\mathcal{L}}^{\otimes m}\right)\otimes k(0)\ \simeq\ H^0(\widetilde{X}, \widetilde{L}^{\otimes m}).\end{align*} $$

As a result, one has that

$$ \begin{align*}V\ =\ \mathscr{V}_0\ \simeq\ \operatorname{\mathrm{Proj}} \bigg(\bigoplus_{m\in {\mathbb N}} H^0(\widetilde{X}, \widetilde{L}^{\otimes m})\bigg)\end{align*} $$

is the ample model of $\widetilde {L}$ on $\widetilde {X}$ , which implies the normality of V and the birationality of $\phi _0$ .

Consider the restriction $\phi |_{\widetilde {\mathscr {Q}}}:\widetilde {\mathscr {Q}}\rightarrow \mathscr {W}$ of $\phi $ to $\widetilde {\mathscr {Q}}$ . As $\mathscr {W}$ is an irreducible scheme over T, whose general fiber is a line, then $\mathscr {W}$ is a surface, and $\mathscr {W}_0$ is also a curve of degree $1$ with respect to $L_V$ , hence a line. On the other hand, if $C\subseteq \widetilde {X}$ is a curve such that $(C.\widetilde {L})=0$ , then $C\subseteq \widetilde {Q}$ because $2\widetilde {L}-(1-\epsilon )\widetilde {Q}$ is ample for $0<\epsilon \ll 1$ . Thus the last statement is proved.

Proof. From the linear equivalence $-K_{\widetilde {X}}\sim 2\widetilde {L}-\widetilde {Q}$ and Lemma 4.10, we know that

$$ \begin{align*}-K_{V}\ =\ (\phi_0)_*(-K_{\widetilde{X}})\ \sim\ 2L_V\end{align*} $$

is an ample $\mathbb {Q}$ -Cartier divisor. Thus V is a ${\mathbb Q}$ -Fano variety as X is klt and $\phi _0^* K_V = K_{\widetilde {X}} - \widetilde {Q}\leq K_{\widetilde {X}}$ .

Next, we show that $2L_V$ is Cartier, which implies that V is a Gorenstein canonical Fano variety. Let $\widetilde {S}\in |-K_{\widetilde {X}}|$ be a general member, and $S_V:=(\phi _0)_* \widetilde {S}$ be its pushforward as a Weil divisor on V. By Theorem 4.3 we know that $(\widetilde {X}, \widetilde {S})$ is a plt log Calabi–Yau pair. Thus $(V,S_V)$ is also a plt log Calabi–Yau pair, which implies that $S_V$ is normal and $\phi _0|_{\widetilde {S}}:\widetilde {S}\to S_V$ is birational. As a result, $(S_V,L_V|_{S_V})$ is a degree $8$ polarized K3 surface as it is the ample model of the quasi-polarized K3 surface $(\widetilde {S},\widetilde {L}|_{\widetilde {S}})$ of degree $8$ . In particular, $L_V|_{S_V}$ is an ample Cartier divisor on $S_V$ , and $\big |2L_V|_{S_V}\big |$ is base-point-free by Theorem 2.2. By Lemma 4.8, we have that

$$ \begin{align*}H^0(V,\mathcal{O}_{V}(2L_V))\ \simeq\ H^0(\widetilde{X},\mathcal{O}_{\widetilde{X}}(2\widetilde{L}))\ \longrightarrow \ H^0(\widetilde{S},\mathcal{O}_{\widetilde{S}}(2\widetilde{L}|_{\widetilde{S}}))\ \simeq \ H^0(S_V,\mathcal{O}_{S_V}(2L_V|_{S_V}))\end{align*} $$

is surjective, and hence $|2L_V|$ is base-point-free. Thus $2L_V$ is Cartier.

Finally we show that $L_V$ is Cartier. By Lemma 4.10 we know that $\phi _0$ induces an isomorphism between $\widetilde {X}\setminus \widetilde {Q}$ and $V\setminus W$ , where $W:=\mathscr {W}_0$ is the central fiber of $W\rightarrow T$ . Since $\widetilde {L}$ is Cartier on $\widetilde {X}$ , it suffices to show that $L_V$ is Cartier near W. As $L_V$ is a $\mathbb {Q}$ -Cartier Weil divisor and V is klt, we know that $\mathcal {O}_{V}(L_V)$ is a Cohen-Macaulay divisorial sheaf by [Reference Kollár and MoriKM98, Corollary 5.25]. Since $S_V\sim 2L_V$ is Cartier, the restriction $\mathcal {O}_{V}(L_V)\otimes _{\mathcal {O}_V}\mathcal {O}_{S_V}$ is also Cohen-Macaulay. Thus one has a natural isomorphism

$$ \begin{align*}\mathcal{O}_{V}(L_V)\otimes_{\mathcal{O}_V}\mathcal{O}_{S_V}\ \simeq\ (\mathcal{O}_{V}(L_V)\otimes_{\mathcal{O}_V}\mathcal{O}_{S_V})^{**} \ \simeq \ \mathcal{O}_{S_V}(L_V|_{S_V}).\end{align*} $$

Notice that $S_V$ contains W. As a consequence, for any point $P\in W$ , one has

$$ \begin{align*}\dim_k(\mathcal{O}_{V}(L_V)\otimes k(P))=\dim_k(\mathcal{O}_{S_V}(L_V|_{S_V})\otimes k(P))=1\end{align*} $$

because $L_V|_{S_V}$ is a Cartier divisor on $S_V$ . Therefore, the sheaf $\mathcal {O}_{V}(L_V)$ is invertible, and hence $L_V$ is Cartier.

Proof. We proved that V is a Gorenstein canonical Fano variety with $-K_V\sim 2L_V$ , where $L_V$ is a Cartier divisor (cf. Proposition 4.11). Moreover, we have $(L_V^3) = (\widetilde {L}^3) = (\widetilde {\mathcal {L}}_t^3) = 4$ for a general $t\in T$ . Hence the statement follows immediately from [Reference FujitaFuj90].

Proof. For a general element $\widetilde {S}\in |-K_{\widetilde {X}}|$ , the image ${S}_V:=\phi _0(\widetilde {S})$ is normal, and hence it is the ample model of $\widetilde {L}|_{\widetilde {S}}$ . In particular, $({S}_V,L_V|_{{S}_V})$ is a degree $8$ polarized K3 surface and hence a $(2,2,2)$ -complete intersection in $\mathbb {P}^5$ . Let $\widetilde {S}_t\in |-K_{\widetilde {\mathscr {X}}_t}|$ be deformations of $\widetilde {S}$ . Since the image of $\widetilde {S}_t$ contains $\mathscr {W}_t$ , then one has that $W\subseteq S_V$ . If W is contained in the singular locus of V, then $S_V$ is double along W, which contradicts the fact that ${S}_V$ is normal.

Proof. We first show that the blow-up is compatible with taking fibers. As both $\widetilde {\mathcal {L}}$ and $-K_{\widetilde {\mathscr {X}}}$ are nef and big over T and $\widetilde {\mathscr {X}}$ is klt, then the Kawamata–Viehweg vanishing theorem implies that

$$ \begin{align*}R^i \widetilde{\pi}_* \widetilde{\mathcal{L}} \ =\ 0\end{align*} $$

for any $i> 0$ and $m\in {\mathbb N}$ . Thus by cohomology and base change, the sheaf $\widetilde {\pi }_* \widetilde {\mathcal {L}}$ is locally free of rank $6$ . By shrinking the base T, we may assume that $T=\operatorname {\mathrm {Spec}} R$ , where R is a DVR with parameter t, and $\mathscr {V}\subseteq \mathbb {P}^5\times T$ . For simplicity, we assume that $\mathscr {W}\subseteq \mathbb {P}^5\times T$ is defined by the ideal $(x_2,...,x_5)$ , and $\mathscr {V}$ is defined locally near the generic point of $\mathscr {W}$ by the ideal $\big (f_1(\underline {x},t),f_2(\underline {x},t)\big )$ . For $i=1,2$ we write

$$ \begin{align*}f_i(\underline{x},t)\ =\ x_0l_{i,0}(\underline{x},t)+x_1l_{i,1}(\underline{x},t)+q_i(\underline{x},t)\end{align*} $$

where $\underline {x}=(x_2,...,x_4)$ , and $l_{i,j}(\underline {x},t)$ (resp. $q_i(\underline {x},t)$ ) is a linear (resp. quadratic) polynomial in variables $\underline {x}$ with coefficients in R. Moreover, by Lemma 4.13 we know that the linear terms (with respect to $\underline {x}$ ) of $f_1(\underline {x},t)$ and $f_2(\underline {x},t)$ are not proportional for any t. In order to prove that $\operatorname {\mathrm {Bl}}_{\mathscr {W}_0}\mathscr {V}_0\simeq (\operatorname {\mathrm {Bl}}_{\mathscr {W}}\mathscr {V})|_{t=0}$ , it suffices to show that the natural morphism

$$ \begin{align*}\frac{(x_2,x_3,x_4,x_5)^k}{t(x_2,...,x_5)^k+(f_1,f_2)} \ \twoheadrightarrow\ \left(\frac{(x_2,x_3,x_4,x_5,t)}{(t,f_1,f_2)}\right)^k\end{align*} $$

is an isomorphism for any $k\gg 0$ . Indeed, for any polynomial $g(\underline {x},t)$ such that the degree of each monomial with respect to $\underline {x}$ is at least k, if $g(\underline {x},0)$ is contained in the ideal generated by $f_i(\underline {x},0)$ for $i=1,2$ , then

$$ \begin{align*}g(\underline{x},0)\ =\ f_1(\underline{x},0)\cdot h_1(\underline{x})+f_2(\underline{x},0)\cdot h_2(\underline{x})\end{align*} $$

for some polynomial h of degree at least $k-1$ . Consequently, one has

$$ \begin{align*}g(\underline{x},t)-f_1(\underline{x},t)\cdot h_1(\underline{x})-f_2(\underline{x},t)\cdot h_2(\underline{x})\ =\ t\cdot p(\underline{x},t)\end{align*} $$

such that the $\underline {x}$ -degree of each monomial in $l(\underline {x},t)$ is at least k as desired.

As an immediate result, we see that $\operatorname {\mathrm {Bl}}_{\mathscr {W}}\mathscr {V}$ is normal. In fact, since $\operatorname {\mathrm {Bl}}_WV=\operatorname {\mathrm {Bl}}_{\mathscr {W}_0}\mathscr {V}_0$ is an integral scheme, then it satisfies (R $_0$ ) and (S $_1$ ) conditions. Thus $\operatorname {\mathrm {Bl}}_{\mathscr {W}}\mathscr {V}$ satisfies (R $_1$ ) and (S $_2$ ) conditions, which is equivalent to being normal.

Notice that we have the desired isomorphism over $T^{\circ }$ . Thus $\operatorname {\mathrm {Bl}}_{\mathscr {W}}\mathscr {V}$ and $\widetilde {\mathscr {X}}$ are birational and isomorphic in codimension one. Let $\mathscr {Q}'$ be the exceptional divisor of the blow-up $\psi :\operatorname {\mathrm {Bl}}_{\mathscr {W}}\mathscr {V}\rightarrow \mathscr {V}$ , and $\mathcal {L}'=\psi ^{*}\phi _* \widetilde {\mathcal {L}}\sim _T \psi ^* \mathcal {O}_{\mathscr {V}}(1)$ . From the blow-up construction we know that the $\mathbb {Q}$ -Cartier ${\mathbb Q}$ -divisor $\mathcal {L}'-\epsilon \mathscr {Q}'$ is ample over T for any rational number $0<\epsilon \ll 1$ . On the other hand, since $\widetilde {\mathcal {L}}$ is nef over T (cf. Proposition 4.9), and $2\widetilde {\mathcal {L}}-(1-\epsilon )\widetilde {\mathscr {Q}}$ is ample over T for any rational number $0<\epsilon \ll 1$ (from the construction of $\widetilde {\mathscr {X}}$ ), we know that $\widetilde {\mathcal {L}}-\epsilon \widetilde {\mathscr {Q}}$ is also ample over T. Since $\operatorname {\mathrm {Bl}}_{\mathscr {W}}\mathscr {V}$ and $\widetilde {\mathscr {X}}$ are isomorphic in codimension $1$ , and $\widetilde {\mathcal {L}}-\epsilon \widetilde {\mathscr {Q}}$ is the birational transform of $\mathcal {L}'-\epsilon \mathscr {Q}'$ , we conclude that $\operatorname {\mathrm {Bl}}_{\mathscr {W}}\mathscr {V}\simeq \widetilde {\mathscr {X}}$ . As a result, we see that $-K_{\widetilde {\mathscr {X}}}$ is ample over T, and hence $\widetilde {\mathscr {X}}\simeq \mathscr {X}$ .

Proof of Theorem 4.1.

It follows from Theorem 4.2(1) that X is Gorenstein canonical. By Corollary 4.12, Lemma 4.13 and Proposition 4.14, we have $X\simeq \operatorname {\mathrm {Bl}}_{\ell }V$ where V is a Gorenstein canonical quartic del Pezzo threefold and $\ell \subseteq V$ is a line. Moreover, X is isomorphic to the blow-up of ${\mathbb P}^3$ along some curve by Lemma 2.4.

Proof. See [Reference Avritzer and LangeAL00, Theorem 4.2] and its proof.

Proof. Let $(\mathcal {P},\ell )$ be a point in W which is of the form in (5.4). Consider the 1-PS $\lambda $ of $\operatorname {\mathrm {SL}}(6)$ of weight $(2,2,-1,-1,-1,-1)$ . The condition $\mu ^t(\mathcal {P},\ell ;\lambda )=2-4t\geq 0$ imposes that $t\leq \frac {1}{2}$ .

Proof. We may assume that $\ell ={\mathbb V}(x_2,x_3,x_4,x_5)$ , V is generated by two quadrics $Q={\mathbb V}(x_0x_2+x_1x_3+q)$ and $Q'={\mathbb V}(x_1x_4+q')$ and that $p=[1,0,...,0]$ is a singularity of V on $\ell $ . Consider the diagonal 1-PS $\lambda $ of $\operatorname {\mathrm {SL}}(6)$ of weight $(1,0,-1,0,0,0)$ , with respect to which one has

$$ \begin{align*}\mu^t(\mathcal{P},\ell;\lambda)=-t<0\end{align*} $$

for any $t>0$ , and thus $(\mathcal {P},\ell )$ is t-GIT unstable.

Proof. Let Q and $Q'$ be two quadrics generating $\mathcal {P}$ . Then one must have that Q and $Q'$ are both reducible and share a common hyperplane. One may thus assume that

$$ \begin{align*}Q={\mathbb V}\big(x_0\ell(\underline{x})\big)\ \ \ \text{and} \ \ Q'={\mathbb V}\big(x_0\ell'(\underline{x})\big).\end{align*} $$

Let $\lambda $ be the 1-PS of $\operatorname {\mathrm {SL}}(6)$ of weight $(-5,1,1,1,1,1)$ . If $(\mathcal {P},\ell )$ is t-GIT semistable, then one has that $\mu ^t(\mathcal {P},\ell ;\lambda )=-8+4t\geq 0$ , that is, $t\geq 2$ , which contradicts Lemma 5.4.

Proof. In [Reference Spotti and SunSS17, Proposition A.2], the authors explicitly construct a 1-PS $\lambda $ such that the corresponding limit $V_0$ has negative GIT weight and is a complete intersection. We claim that the t-GIT weight of the pair $(V,\ell )$ under $\lambda $ is also negative, for any $\ell $ and any $t<\frac {5}{26}$ .

A classical result of Weierstrass and Segre (cf. [Reference Avritzer and LangeAL00]) allows one to put a pencil $\mathcal {P}$ of quadrics into a canonical form $\mathcal {P} = uQ + vQ'$ , where Q and $Q'$ are block diagonal. A systematic treatment can be found in [Reference Hodge and PedoeHP94, pp. 286–292]. Let $r_0$ denote the number of variables that do not appear in $\mathcal {P}$ . The discriminant $\Delta (\mathcal {P})$ defines the subscheme of ${\mathbb P}^1$ parameterizing singular quadrics in the pencil. In [Reference Hodge and PedoeHP94, pp.289, Eqns.(1)–(2)], two matrices A and B are introduced which play a role in the classification. For our purposes, since our quadrics are $4$ -dimensional, there are only a few cases to consider, so we do not define them here; instead, we write down the explicit equations of the quadrics as needed.

Nondegenerate case This is the case where $\Delta (\mathcal {P})$ is not identically zero. If $\mathcal {P}$ is non-degenerate and GIT unstable, then $\Delta (\mathcal {P})$ has a root of multiplicity $\geq 4$ . A general pencil satisfying this has Segre symbol

1. (1) either $[4,1,1]$ , that is, it is defined by the equations

   $$ \begin{align*}\begin{cases} x_0x_3+x_1x_2+x_4^2+x_5^2 =0 \\ x_1x_3+x^2_2+ax_4^2+bx_5^2 =0 \end{cases}\!\!\!\!\!,\end{align*} $$
   where $a,b\neq 1$ are two different numbers.

2. (2) or $[4,2]$ , that is, it is defined by the equations

   $$ \begin{align*}\begin{cases} x_0x_3+x_1x_2+x_5^2+2ax_4x_5 =0 \\ x_1x_3+x^2_2+2x_4x_5 =0 \end{cases}\!\!\!\!\!\!,\end{align*} $$
   where $a\neq 1$ is a complex number.

In both cases, one can take the diagonal 1-PS $\lambda $ of weight $(5,2,-1,-4,-1,-1)$ . With respect to this weight, one has that

$$ \begin{align*}\mu(\mathcal{P},\lambda)=\max\{1,-2\}-2=-1, \ \ \text{and} \ \ \mu([\ell],\lambda)\leq -(-4-1)=5,\end{align*} $$

and thus

$$ \begin{align*}\mu^{t}(\mathcal{P},\ell;\lambda)\ \leq \ -1+ 5t\ <\ 0\end{align*} $$

for any line $\ell \subseteq \operatorname {\mathrm {Bs}}(\mathcal {P})$ and any rational number $0\leq t<\frac {5}{26}$ .

Pure degenerate case $\mathbf {r_0=0}$ and $\mathbf {A=B=0}$ There is only one case to deal with, that is, when $Q={\mathbb V}(x_0x_1+x_2x_3)$ and $Q'={\mathbb V}(x_1x_2+x_3x_4)$ . One can take the 1-PS $\lambda $ of weight $(2,-2,1,-1,0,0)$ , with respect to which one has

$$ \begin{align*}\mu^{t}(\mathcal{P},\ell;\lambda)\ \leq \ -1+\frac{1}{5}\cdot 3\ <\ 0.\end{align*} $$

Mixed general case $\mathbf {r_0 = 0}$ , but $\mathbf {A,B \neq 0}$ There is only one case to deal with, that is, when $Q={\mathbb V}(x_0x_1+x_2x_3+x_5^2)$ and $Q'={\mathbb V}(x_1x_2+x_3x_4)$ . One can take the 1-PS $\lambda $ of weight $(1,-3,3,-5,5,-1)$ , with respect to which one has

$$ \begin{align*}\mu^{t}(\mathcal{P},\ell;\lambda)\ <\ -2+\frac{1}{5}\cdot 8\ <\ 0.\end{align*} $$

Case $\mathbf {r_0\neq 0}$ in the canonical form This is an easy case: $\mathcal {P}$ has nonfinite isotropy group and one can explicitly write down a destabilizing 1-PS that fixes $\mathcal {P}$ with negative GIT weight, because one has the freedom to choose weight for the last $r_0$ variables. For example, suppose that no $x_0$ appears in the defining equations of the quadrics in the pencil $\mathcal {P}$ . Then one may take the $1$ -PS with weights $(5m,-m,-m,-m,-m,-m)$ for some $m>0$ . In this case, the Hilbert–Mumford weight satisfies

$$\begin{align*}\mu^{t}(\mathcal{P},\ell;\lambda)\ \le\ -4m+2mt \,<\, 0 \qquad\text{for any } t<1.\\[-37pt] \end{align*}$$

Proof. If V is K-unstable, then by [Reference Spotti and SunSS17, Proposition A.2], one can find an explicit diagonal 1-PS $\lambda $ of $\operatorname {\mathrm {SL}}(6)$ such that the limit $V_0:=\lim _{t\to 0}\lambda (t)\cdot V$ is also a complete intersection of two quadrics, and $\mu (\mathcal {P};\lambda )<0$ . In fact, using the same 1-PS exhibited in [Reference Spotti and SunSS17, Proposition A.2], one can show that $\mu ^{t_0}(\mathcal {P},\ell ';\lambda )<0$ for any line $\ell '\subseteq V_0$ ; see Proposition 5.7. Let $\mathscr {V}\rightarrow {\mathbb A}^1$ be the test configuration of V induced by $\lambda $ , let $\mathscr {L}$ be the closure of $\ell \times ({\mathbb A}^1\setminus \{0\})$ in $\mathscr {V}$ , and $\ell _0:=\mathscr {L}\cap V_0$ be the limit of $\ell $ under this degeneration, which is also a line. Then $\mathscr {X}:=\operatorname {\mathrm {Bl}}_{\mathscr {L}}\mathscr {V}\rightarrow {\mathbb A}^1$ is a test configuration of $X=\operatorname {\mathrm {Bl}}_{\ell }V$ . If X is K-semistable, then one has

$$ \begin{align*}\mu^{t_0}(V,\ell;\lambda) \ =\ c\cdot \mu^{\lambda_{\operatorname{\mathrm{CM}}}}(\mathscr{X}_0;\lambda) \ = \ c'\cdot \operatorname{\mathrm{Fut}}(\mathscr{X}) \ \geq\ 0,\end{align*} $$

where $c,c'>0$ are two constants. This leads to a contradiction, and hence (1) holds true. This argument also shows that there is no GIT walls in $(0,t_0]$ .

Suppose now X is K-semistable but $(V,\ell )$ is $t_0$ -GIT unstable. Then by the structure of variation of GIT, there exists a 1-PS $\lambda $ of $\operatorname {\mathrm {SL}}(6)$ such that $\mu ^0(V,\ell ;\lambda )=0$ and $\mu ^t(V,\ell ;\lambda )<0$ for any $t>0$ . Since V is GIT semistable by (1), then the limit $V_0:=\lim _{s\to 0}\lambda (s)\cdot V$ is also GIT semistable, and hence is a $(2,2)$ -complete intersection. Let $\mathscr {V}\rightarrow {\mathbb A}^1$ be the test configuration of V induced by $\lambda $ , let $\mathscr {L}$ be the closure of $\ell \times ({\mathbb A}^1\setminus \{0\})$ in $\mathscr {V}$ , and $\ell _0:=\mathscr {L}\cap V_0$ be the limit of $\ell $ under this degeneration, which is also a line. Then $\mathscr {X}:=\operatorname {\mathrm {Bl}}_{\mathscr {L}}\mathscr {V}\rightarrow {\mathbb A}^1$ is a test configuration of $X=\operatorname {\mathrm {Bl}}_{\ell }V$ . Since X is K-semistable, then one has

$$ \begin{align*}0\>\ \mu^{t_0}(V,\ell;\lambda) \ =\ c\cdot \mu^{\lambda_{\operatorname{\mathrm{CM}}}}(\mathscr{X}_0;\lambda) \ = \ c'\cdot \operatorname{\mathrm{Fut}}(\mathscr{X}) \ \geq\ 0,\end{align*} $$

which is a contradiction. The K-(poly)stability part in statement (2) is similar.

The last statement follows from Lemma 5.5.

Proof. This follows immediately from Proposition 6.2, Lemma 5.5 and the paragraph above it.

Proof. Let C be a $(-K_X)$ -line, that is, a smooth rational curve C such that $(C.-K_X)=1$ . Let F be the exceptional divisor of the blow-up $\phi :X\rightarrow V$ , and E be the exceptional divisor of the contraction $\pi :X\rightarrow {\mathbb P}^3$ . Set $H:=\pi ^{*}\mathcal {O}_{{\mathbb P}^3}(1)$ and $L:=\pi ^{*}\mathcal {O}_{V}(1)$ to be two big and nef divisors. Then one has that $-K_X\sim L+H$ , and that $(C.L+H)=1$ . Thus either $(C.H)=0$ , in which case C is contained in E, or $(C.L)=0$ , in which case C is contained in F. Moreover, since $H+L$ is ample, these two cases cannot occur simultaneously. Notice that the lines contained in F is parametrized by $\ell $ (a smooth rational curve), while the lines contained in E is parametrized by a curve which is either irrational or singular.

Proof. By the universal property of blow-ups, one can lift an automorphism of V preserving $\ell $ to an automorphism of X, and hence there is a natural injection $\operatorname {\mathrm {Aut}}(V,\ell )\hookrightarrow \operatorname {\mathrm {Aut}}(X)$ . Conversely, if $g\in \operatorname {\mathrm {Aut}}(X)$ is an automorphism of X, then g must preserve the exceptional divisor F of $X\rightarrow V$ by Lemma 6.4. The rest of the proof is identical to [Reference Liu and ZhaoLZ25, Lemma 5.8].

Proof. Since $U^K$ parametrizes K-semistable $\mathbb {Q}$ -Fano varieties, then by universality of K-moduli stacks, there exists a morphism

To show $\varphi $ is an isomorphism, we will construct the its inverse morphism. By the construction of K-moduli stacks, we know that

where $T\subseteq {\mathrm {Hilb}}_{\chi }(\mathbb {P}^{N})$ is a locally closed subscheme of the Hilbert scheme with respect to the Hilbert polynomial $\chi (m):=\chi (X,-mrK_X)$ for $r\in {\mathbb Z}_{>0}$ sufficiently divisible. By Corollary 6.11, we know that T is irreducible and smooth. Let $\pi :\mathscr {Z}\rightarrow T$ be the universal family. By Lemma 6.4, there is a unique divisor $\mathscr {F}\subseteq \mathscr {Z}$ relative Cartier over T such that each fiber $\mathscr {F}_t$ is the exceptional divisor of the blow-up along a line.

Consider the $\pi $ -big and $\pi $ -nef line bundle $\mathscr {R}:=-\omega _{\mathscr {Z}/T}+\mathscr {F}$ on $\mathscr {Z}$ , whose restriction on each fiber $\mathscr {Z}_t$ is isomorphic to the pull-back of the line bundle $\mathcal {O}_{\mathbb {P}^5}(2)$ . Taking the $\pi $ -ample model with respect to $\mathscr {R}$ , one obtains a family $\mathscr {V}\rightarrow T$ such that each fiber $\mathscr {V}_t$ is isomorphic a $(2,2)$ -complete intersection in ${\mathbb P}^5$ .

Take a fppf cover $\sqcup _i T_i\rightarrow T$ of T such that the pull-back family $p_i:\mathscr {V}_i\rightarrow T_i$ can be embedded into a trivial $\mathbb {P}^5$ -bundle over $T_i$ . Let $\mathcal {L}=\sqcup _i \mathcal {L_i}$ be the line bundle on $\sqcup _i \mathscr {V}_i$ which is isomorphic to the pull-back of $\mathcal {O}_{\mathbb {P}^5}(1)$ over each $T_i$ . By Kawamata-Viehweg vanishing, we know that $(p_i)_*\mathcal {L}_i$ is a rank $6$ vector bundle over $T_i$ . Let $\mathcal {P}_i/T_i$ be the $\operatorname {\mathrm {PGL}}(6)$ -torsor induced by projectivized basis of $(p_i)_*\mathcal {L}_i$ . As the cocycle condition of $\{(p_i)_*\mathcal {L}_i/T_i\}_i$ is off by $\pm 1$ , then we deduce that $\{\mathcal {P}_i/T_i\}_i$ is a fppf descent datum descending to a $\operatorname {\mathrm {PGL}}(6)$ -torsor $\mathcal {P}/T$ , which is $\operatorname {\mathrm {PGL}}(N+1)$ -equivariant. This induces a morphism

Since both stacks are smooth, then $\varphi ^{-1}$ is indeed the inverse of $\varphi $ .

Proof. Let $(V,\ell )\in W$ be a $t_0$ -GIT semistable pair. Pick a family of pairs $\{(V_t,\ell _t)\}_{t\in T}$ over a smooth pointed curve $(0 \in T)$ such that $(V_0,\ell _0)=(V,\ell )$ , and $\operatorname {\mathrm {Bl}}_{\ell _t} V_t$ is K-stable and smooth for any $t\in T\setminus \{0\}$ . After possibly a base change, there exists a K-polystable filling $X'$ by the properness of K-moduli spaces, which is isomorphic to the blow-up of a quartic del Pezzo threefold $V'$ along a line $\ell '$ . By the separatedness of GIT quotients, we know that $(V,\ell )$ admits a special degeneration to $g \cdot (V',\ell ')$ for some $g \in \operatorname {\mathrm {PGL}}(6)$ . Thus $X=\operatorname {\mathrm {Bl}}_{\ell }V$ is K-semistable by openness of K-semistability. Suppose in addition that $(V,\ell )$ is GIT polystable. Let $X_0':=\operatorname {\mathrm {Bl}}_{\ell ^{\prime }_0}V_0'$ be the K-polystable limit of X, for which we know that $(V_0',\ell _0')$ is GIT polystable by Proposition 6.2, and hence is S-equivalent to $(V,\ell )$ . Therefore one has $(V,\ell )= g \cdot (V_0',\ell _0')$ for some $g \in \operatorname {\mathrm {PGL}}(6)$ and X is K-polystable.