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The answer is yes, in fact the following result holds.

Theorem. Let $f \colon X \to T$ be a flat deformation of a Fano variety $X_0:=f^{-1}(0)$ having at most terminal, $\mathbb{Q}$-factorial singularities.

 

Then $X_t:=f^{-1}(t)$ is a Fano variety with at most terminal, $\mathbb{Q}$-factorial singularities, for all $t$ in a neighborhood of $0$ in $T$.

For a proof, see Corollary 3.2 and Proposition 3.8 in

De Fernex, Tommaso; Hacon, Christopher D., Deformations of canonical pairs and Fano varieties, J. Reine Angew. Math. 651, 97-126 (2011). ZBL1220.14026.

The answer is yes, in fact the following result holds.

Theorem. Let $f \colon X \to T$ be a flat deformation of a Fano variety $X_0:=f^{-1}(0)$ having at most terminal, $\mathbb{Q}$-factorial singularities.

 

Then $X_t:=f^{-1}(t)$ is a Fano variety with at most terminal, $\mathbb{Q}$-factorial singularities, for all $t$ in a neighborhood of $0$ in $T$.

For a proof, see Corollary 3.2 and Proposition 3.8 in

De Fernex, Tommaso; Hacon, Christopher D., Deformations of canonical pairs and Fano varieties, J. Reine Angew. Math. 651, 97-126 (2011). ZBL1220.14026.

The answer is yes, in fact the following result holds.

Theorem. Let $f \colon X \to T$ be a flat deformation of a Fano variety $X_0:=f^{-1}(0)$ having at most terminal, $\mathbb{Q}$-factorial singularities.

Then $X_t:=f^{-1}(t)$ is a Fano variety with at most terminal, $\mathbb{Q}$-factorial singularities, for all $t$ in a neighborhood of $0$ in $T$.

For a proof, see Corollary 3.2 and Proposition 3.8 in

De Fernex, Tommaso; Hacon, Christopher D., Deformations of canonical pairs and Fano varieties, J. Reine Angew. Math. 651, 97-126 (2011). ZBL1220.14026.

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Francesco Polizzi
Francesco Polizzi
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The answer is yes, in fact the following result holds.

Theorem. Let $f \colon X \to T$ be a flat deformation of a Fano variety $X_0:=f^{-1}(0)$, having at most terminal, $\mathbb{Q}$-factorial singularities.

Then $X_t:=f^{-1}(t)$ is a Fano variety with at most terminal, $\mathbb{Q}$-factorial singularities, for all $t$ in a neighborhood of $0$ in $T$.

For a proof, see Corollary 3.2 and Proposition 3.8 in

De Fernex, Tommaso; Hacon, Christopher D., Deformations of canonical pairs and Fano varieties, J. Reine Angew. Math. 651, 97-126 (2011). ZBL1220.14026.

The answer is yes, in fact the following result holds.

Theorem. Let $f \colon X \to T$ be a flat deformation of a Fano variety $X_0:=f^{-1}(0)$, having at most terminal, $\mathbb{Q}$-factorial singularities.

Then $X_t:=f^{-1}(t)$ is a Fano variety with at most terminal, $\mathbb{Q}$-factorial singularities, for all $t$ in a neighborhood of $0$ in $T$.

For a proof, see Corollary 3.2 and Proposition 3.8 in

De Fernex, Tommaso; Hacon, Christopher D., Deformations of canonical pairs and Fano varieties, J. Reine Angew. Math. 651, 97-126 (2011). ZBL1220.14026.

The answer is yes, in fact the following result holds.

Theorem. Let $f \colon X \to T$ be a flat deformation of a Fano variety $X_0:=f^{-1}(0)$ having at most terminal, $\mathbb{Q}$-factorial singularities.

Then $X_t:=f^{-1}(t)$ is a Fano variety with at most terminal, $\mathbb{Q}$-factorial singularities, for all $t$ in a neighborhood of $0$ in $T$.

For a proof, see Corollary 3.2 and Proposition 3.8 in

De Fernex, Tommaso; Hacon, Christopher D., Deformations of canonical pairs and Fano varieties, J. Reine Angew. Math. 651, 97-126 (2011). ZBL1220.14026.

Source Link
Francesco Polizzi
Francesco Polizzi
  • 66.3k
  • 5
  • 180
  • 283

The answer is yes, in fact the following result holds.

Theorem. Let $f \colon X \to T$ be a flat deformation of a Fano variety $X_0:=f^{-1}(0)$, having at most terminal, $\mathbb{Q}$-factorial singularities.

Then $X_t:=f^{-1}(t)$ is a Fano variety with at most terminal, $\mathbb{Q}$-factorial singularities, for all $t$ in a neighborhood of $0$ in $T$.

For a proof, see Corollary 3.2 and Proposition 3.8 in

De Fernex, Tommaso; Hacon, Christopher D., Deformations of canonical pairs and Fano varieties, J. Reine Angew. Math. 651, 97-126 (2011). ZBL1220.14026.