A cone would fit the given description
A cone has a curved base edge and 1 vertex
A shape with 1 vertex, 1 edge, 1 flat face, and 1 curved face is a cone. The vertex is the tip of the cone, the edge is the circular base where the flat face meets the curved surface, and the flat face is the circular base itself. The curved face extends from the base to the vertex.
One curved surface (two if it is hollow) and no edges.
A sphere !
A hemisphere as two faces (one curved and one plane), one edge and no vertices.
A cone would fit the given description
It has a flat base, a curved surface area, a circular edge and one vertex.
a cone
A cone has a curved base edge and 1 vertex
4 faces and 2 vertices.
One curved surface (two if it is hollow) and no edges.
A sphere !
A hemisphere as two faces (one curved and one plane), one edge and no vertices.
An hemisphere is half of a globe with a circular flat base, a round edge, a curved surface but no vertices.
A hemisphere as two faces (one curved and one plane), one edge and no vertices.
A cylinder would fit the given description
A finite cone has one vertex (not vertices!), 1 edge and 2 faces.