If you filed for child support, you are at least asserting--though not confirming--that he is the father, so he has the right to file for timeshare ("custody"). Actually, he has that right whether you file for child support or not. If there is a very serious reason why the father should not be alone with the child (like maybe he is a child molester), then you may be able to convince the judge that the father should have supervised visits, but the reason has to be extremely serious. Your background may sway the judge, but it depends on what your conviction was for. If you and the father argue over your son's last name, the judge will likely hyphenate your two last names. You have the right to restrict the father's visits, but talk to a lawyer before you do. restricting visits can be used against you as it will look like you can't act as a co-parent.
The contents of this answer should be considered friendly advice, not legal advice (I'm a pretty friendly guy), and the answer should not be construed to constitute an attorney-client relationship. If you'd like actual legal advice, call me for a free consultation at 813-635-0222. Also, if you liked this answer as much as my big ego thinks you did, be sure to click the thumbs-up button!
I agree that, to some extent, you seem to be attempting a "double-dip". On one hand, you want the benefits (e.g., child support payments) of this man being the father, but you don't want the burdens (e.g., allowing the man to spend time with you son) of him being the father.
Did you file for child support with or without an attorney? A local attorney would be able to help you navigate the web of rules and laws that govern family and domestic relations. It would likely be a good investment for you, since there's so much at stake.
Best of luck.
Andrew M. Bonderud, Esq.
The Bonderud Law Firm, P.A.
Andrew M. Bonderud, Esq. is an attorney The Bonderud Law Firm, P.A. He offers free consultations 24/7. You can reach him at 904-438-8082. Andrew's posting here is not to be considered legal advice nor does an attorney-client relationship exist.
Yes the father can file for custody and yes a criminal history can have an affect on the outcome of the case. Consult with an attorney, and before you do, read the provisions of Florida Statutes 61.13(3). These are the factors the court uses in determining what should be done.
R. Jason de Groot, Esq., 386-337-8239
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