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In this paper, we investigate the sectional contravariant curvature of a doubly warped product manifold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mrow><mo> </mo></mrow><mi>f</mi></msub><mi>B</mi><msub><mo>×</mo><mi>b</mi></msub><mi>F</mi><mo>,</mo><mover accent="true"><mi>g</mi><mo stretchy="false">˜</mo></mover><mo>,</mo><mi mathvariant="normal">Π</mi><mo>=</mo><msub><mi mathvariant="normal">Π</mi><mi>B</mi></msub><mo>+</mo><msub><mi mathvariant="normal">Π</mi><mi>F</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> equipped with a product Poisson structure <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">Π</mi></semantics></math></inline-formula>, using warping functions and sectional curvatures of its factor manifolds <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>B</mi><mo>,</mo><msub><mover accent="true"><mi>g</mi><mo stretchy="false">˜</mo></mover><mi>B</mi></msub><mo>,</mo><msub><mi mathvariant="normal">Π</mi><mi>B</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>F</mi><mo>,</mo><msub><mover accent="true"><mi>g</mi><mo stretchy="false">˜</mo></mover><mi>F</mi></msub><mo>,</mo><msub><mi mathvariant="normal">Π</mi><mi>F</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula>. Qualar and null sectional contravariant curvatures of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mrow><mo> </mo></mrow><mi>f</mi></msub><mi>B</mi><msub><mo>×</mo><mi>b</mi></msub><mi>F</mi><mo>,</mo><mover accent="true"><mi>g</mi><mo stretchy="false">˜</mo></mover><mo>,</mo><mi mathvariant="normal">Π</mi><mo>)</mo></mrow></semantics></math></inline-formula> are also given. As an example, we construct a four-dimensional Lorentzian doubly warped product Poisson manifold where qualar and sectional curvatures are obtained.