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We work over the complex numbers. Fix a genus g. Does there exist a connected reduced base $ B $ and a flat projective family $ \pi : X \rightarrow B $ satisfying the following two conditions?

(i) its general fiber is a nonsingular curve of genus $ g $. (ii) it has a special fiber $ X_0 $ which is an irreducible, projective one-dimensional scheme such that $ (X_0)_{red} $ is isomorphic to $ \mathbb{P}^1$.

Toy cases: If $ g $ is a triangular number, write $ g = \frac{(d-1)(d-2)}{2} $ and we can take the universal family of degree $ d $ hypersurfaces in $ \mathbb{P}^2 $. If $ g = 4 $, we can take the family of complete intersections of type $ (2,3) $ in $ \mathbb{P}^3 $. This family has a fiber whose reduction is the twisted cubic. Similarly, if $ g=5 $, complete intersections of type $ (2,2,2) $ in $ \mathbb{P}^4 $ work. I don't know what happens for genera which can never occur as genera of complete intersection curves in projective space, say $ g=2,7,8 \ldots $, although I think these occur as intersections in other varieties such as Grassmanians / products of projective spaces.

We work over the complex numbers. Fix a genus $g$. Does there exist a connected reduced base $ B $ and a flat projective family $ \pi : X \rightarrow B $ satisfying the following two conditions?

1. its general fiber is a nonsingular curve of genus $ g $.
2. it has a special fiber $ X_0 $ which is an irreducible, projective one-dimensional scheme such that $ (X_0)_\text{red} $ is isomorphic to $ \mathbb{P}^1$.

Toy cases: If $ g $ is a triangular number, write $ g = \frac{(d-1)(d-2)}{2} $ and we can take the universal family of degree $ d $ hypersurfaces in $ \mathbb{P}^2 $. If $ g = 4 $, we can take the family of complete intersections of type $ (2,3) $ in $ \mathbb{P}^3 $. This family has a fiber whose reduction is the twisted cubic. Similarly, if $ g=5 $, complete intersections of type $ (2,2,2) $ in $ \mathbb{P}^4 $ work. I don't know what happens for genera which can never occur as genera of complete intersection curves in projective space, say $ g=2,7,8 \dotsc $, although I think these occur as intersections in other varieties such as Grassmanians / products of projective spaces.

We work over the complex numbers. Fix a genus g. Does there exist a reduced base $ B $ and a flat projective family $ \pi : X \rightarrow B $ satisfying the following two conditions?

(i) its general fiber is a nonsingular curve of genus $ g $. (ii) it has a special fiber $ X_0 $ which is an irreducible, projective one-dimensional scheme such that $ (X_0)_{red} $ is isomorphic to $ \mathbb{P}^1$.

Toy cases: If $ g $ is a triangular number, write $ g = \frac{(d-1)(d-2)}{2} $ and we can take the universal family of degree $ d $ hypersurfaces in $ \mathbb{P}^2 $. If $ g = 4 $, we can take the family of complete intersections of type $ (2,3) $ in $ \mathbb{P}^3 $. This family has a fiber whose reduction is the twisted cubic. Similarly, if $ g=5 $, complete intersections of type $ (2,2,2) $ in $ \mathbb{P}^4 $ work. I don't know what happens for genera which can never occur as genera of complete intersection curves in projective space, say $ g=2,7,8 \ldots $, although I think these occur as intersections in other varieties such as Grassmanians / products of projective spaces.

We work over the complex numbers. Fix a genus g. Does there exist a connected reduced base $ B $ and a flat projective family $ \pi : X \rightarrow B $ satisfying the following two conditions?

(i) its general fiber is a nonsingular curve of genus $ g $. (ii) it has a special fiber $ X_0 $ which is an irreducible, projective one-dimensional scheme such that $ (X_0)_{red} $ is isomorphic to $ \mathbb{P}^1$.

Toy cases: If $ g $ is a triangular number, write $ g = \frac{(d-1)(d-2)}{2} $ and we can take the universal family of degree $ d $ hypersurfaces in $ \mathbb{P}^2 $. If $ g = 4 $, we can take the family of complete intersections of type $ (2,3) $ in $ \mathbb{P}^3 $. This family has a fiber whose reduction is the twisted cubic. Similarly, if $ g=5 $, complete intersections of type $ (2,2,2) $ in $ \mathbb{P}^4 $ work. I don't know what happens for genera which can never occur as genera of complete intersection curves in projective space, say $ g=2,7,8 \ldots $, although I think these occur as intersections in other varieties such as Grassmanians / products of projective spaces.

We work over the complex numbers. Fix a genus g. Does there exist a reduced base $ B $ and a flat projective family $ \pi : X \rightarrow B $ satisfying the following two conditions?

(i) its general fiber is a nonsingular curve of genus $ g $. (ii) it has a special fiber $ X_0 $ which is an irreducible, projective one-dimensional scheme such that $ (X_0)_{red} $ is isomorphic to $ \mathbb{P}^1$.

We work over the complex numbers. Fix a genus g. Does there exist a reduced base $ B $ and a flat projective family $ \pi : X \rightarrow B $ satisfying the following two conditions?

(i) its general fiber is a nonsingular curve of genus $ g $. (ii) it has a special fiber $ X_0 $ which is an irreducible, projective one-dimensional scheme such that $ (X_0)_{red} $ is isomorphic to $ \mathbb{P}^1$.

Toy cases: If $ g $ is a triangular number, write $ g = \frac{(d-1)(d-2)}{2} $ and we can take the universal family of degree $ d $ hypersurfaces in $ \mathbb{P}^2 $. If $ g = 4 $, we can take the family of complete intersections of type $ (2,3) $ in $ \mathbb{P}^3 $. This family has a fiber whose reduction is the twisted cubic. Similarly, if $ g=5 $, complete intersections of type $ (2,2,2) $ in $ \mathbb{P}^4 $ work. I don't know what happens for genera which can never occur as genera of complete intersection curves in projective space, say $ g=2,7,8 \ldots $, although I think these occur as intersections in other varieties such as Grassmanians / products of projective spaces.